Lemma 55.6.2 (Genus one). The minimal numerical types of genus one are up to equivalence

1. $n = 1$, $a_{11} = 0$, $g_1 = 1$, $m_1, w_1 \geq 1$ arbitrary,

2. $n = 2$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w \\ 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

3. $n = 2$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 4w \\ 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

4. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w \\ w & -2w & w \\ w & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

5. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 \\ w & -2w & 3w \\ 0 & 3w & -6w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ 3w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

6. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 3m \end{matrix} \right), \quad \left( \begin{matrix} -6w & 3w & 0 \\ 3w & -6w & 3w \\ 0 & 3w & -2w \end{matrix} \right), \quad \left( \begin{matrix} 3w \\ 3w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

7. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 \\ 2w & -4w & 4w \\ 0 & 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

8. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 \\ 2w & -4w & 2w \\ 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

9. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 \\ 2w & -2w & 2w \\ 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

10. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & w \\ w & -2w & w & 0 \\ 0 & w & -2w & w \\ w & 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

11. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 \\ 2w & -4w & 2w & 0 \\ 0 & 2w & -4w & 4w \\ 0 & 0 & 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

12. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 \\ 2w & -4w & 2w & 0 \\ 0 & 2w & -4w & 2w \\ 0 & 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

13. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 \\ 2w & -2w & w & 0 \\ 0 & w & -2w & 2w \\ 0 & 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

14. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} 2m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w & 2w \\ w & -2w & 0 & 0 \\ w & 0 & -2w & 0 \\ 2w & 0 & 0 & -4w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

15. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} 2m \\ m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 2w & 2w \\ 2w & -4w & 0 & 0 \\ 2w & 0 & -4w & 0 \\ 2w & 0 & 0 & -2w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

16. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & w \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & 0 \\ 0 & 0 & w & -2w & w \\ w & 0 & 0 & w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

17. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 3m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & 2w & 0 \\ 0 & 0 & 2w & -4w & 2w \\ 0 & 0 & 0 & 2w & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ 2w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

18. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -2w & w \\ 0 & 0 & 0 & w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ 2w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

19. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} 2m \\ 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -4w & 4w \\ 0 & 0 & 0 & 4w & -8w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

20. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -4w & 2w \\ 0 & 0 & 0 & 2w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

21. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & 0 \\ 0 & 0 & w & -2w & 2w \\ 0 & 0 & 0 & 2w & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

22. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} 2m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w & w & w \\ w & -2w & 0 & 0 & 0 \\ w & 0 & -2w & 0 & 0 \\ w & 0 & 0 & -2w & 0 \\ w & 0 & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

23. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 2m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & w \\ 0 & 0 & w & -2w & 0 \\ 0 & 0 & w & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

24. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} 2m \\ 2m \\ 2m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 2w \\ 0 & 0 & 2w & -4w & 0 \\ 0 & 0 & 2w & 0 & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

25. $n \geq 6$ and we have an $n$-cycle generalizing (16):

1. $m_1 = \ldots = m_ n = m$,

2. $a_{12} = \ldots = a_{(n - 1) n} = w$, $a_{1n} = w$, and for other $i < j$ we have $a_{ij} = 0$,

3. $w_1 = \ldots = w_ n = w$

with $w$ and $m$ arbitrary,

26. $n \geq 6$ and we have a chain generalizing (19):

1. $m_1 = \ldots = m_{n - 1} = 2m$, $m_ n = m$,

2. $a_{12} = \ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 1) n} = 4w$, and for other $i < j$ we have $a_{ij} = 0$,

3. $w_1 = w$, $w_2 = \ldots = w_{n - 1} = 2w$, $w_ n = 4w$

with $w$ and $m$ arbitrary,

27. $n \geq 6$ and we have a chain generalizing (20):

1. $m_1 = \ldots = m_ n = m$,

2. $a_{12} = \ldots = a_{(n - 1) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,

3. $w_1 = w$, $w_2 = \ldots = w_{n - 1} = 2w$, $w_ n = w$

with $w$ and $m$ arbitrary,

28. $n \geq 6$ and we have a chain generalizing (21):

1. $m_1 = w$, $w_2 = \ldots = m_{n - 1} = 2m$, $m_ n = m$,

2. $a_{12} = 2w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 1) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$,

3. $w_1 = 2w$, $w_2 = \ldots = w_{n - 1} = w$, $w_ n = 2w$

with $w$ and $m$ arbitrary,

29. $n \geq 6$ and we have a type generalizing (23):

1. $m_1 = m$, $m_2 = \ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_ n = m$,

2. $a_{12} = 2w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,

3. $w_1 = 2w$, $w_2 = \ldots = w_ n = w$

with $w$ and $m$ arbitrary,

30. $n \geq 6$ and we have a type generalizing (24):

1. $m_1 = \ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_ n = m$,

2. $a_{12} = \ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 2) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$,

3. $w_1 = w$, $w_2 = \ldots = w_ n = 2w$

with $w$ and $m$ arbitrary,

31. $n \geq 6$ and we have a type generalizing (22):

1. $m_1 = m_2 = m$, $m_3 = \ldots = m_{n - 2} = 2m$, $m_{n - 1} = m_ n = m$,

2. $a_{13} = w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,

3. $w_1 = \ldots = w_ n = w$,

with $w$ and $m$ arbitrary,

32. $n = 7$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 3m \\ m \\ 2m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & w & -2w & 0 & w & 0 & w \\ 0 & 0 & 0 & -2w & w & 0 & 0 \\ 0 & 0 & w & w & -2w & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -2w & w \\ 0 & 0 & w & 0 & 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

33. $n = 8$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 3m \\ 2m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & 0 & w & -2w & w & 0 & 0 & w \\ 0 & 0 & 0 & w & -2w & w & 0 & 0 \\ 0 & 0 & 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & 0 & w & 0 & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary,

34. $n = 9$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

$\left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 5m \\ 6m \\ 4m \\ 2m \\ 3m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & w & -2w & w & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & w & -2w & w & 0 & w \\ 0 & 0 & 0 & 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & 0 & 0 & 0 & w & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right)$

with $w$ and $m$ arbitrary.

Proof. This is proved in Section 55.5. See discussion at the start of Section 55.5. $\square$

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