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Properties

Label 40.960.65.ir.1
Level $40$
Index $960$
Genus $65$
Analytic rank $12$
Cusps $32$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $40$ Newform level: $1600$
Index: $960$ $\PSL_2$-index:$960$
Genus: $65 = 1 + \frac{ 960 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$
Cusps: $32$ (none of which are rational) Cusp widths $20^{16}\cdot40^{16}$ Cusp orbits $4^{4}\cdot8^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $12$
$\Q$-gonality: $11 \le \gamma \le 20$
$\overline{\Q}$-gonality: $11 \le \gamma \le 20$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.960.65.328

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}7&0\\10&37\end{bmatrix}$, $\begin{bmatrix}17&20\\20&37\end{bmatrix}$, $\begin{bmatrix}21&32\\12&17\end{bmatrix}$, $\begin{bmatrix}21&32\\22&27\end{bmatrix}$, $\begin{bmatrix}25&8\\12&25\end{bmatrix}$, $\begin{bmatrix}29&8\\18&3\end{bmatrix}$
$\GL_2(\Z/40\Z)$-subgroup: $C_{24}:C_2^5$
Contains $-I$: yes
Quadratic refinements: 40.1920.65-40.ir.1.1, 40.1920.65-40.ir.1.2, 40.1920.65-40.ir.1.3, 40.1920.65-40.ir.1.4, 40.1920.65-40.ir.1.5, 40.1920.65-40.ir.1.6, 40.1920.65-40.ir.1.7, 40.1920.65-40.ir.1.8, 40.1920.65-40.ir.1.9, 40.1920.65-40.ir.1.10, 40.1920.65-40.ir.1.11, 40.1920.65-40.ir.1.12, 40.1920.65-40.ir.1.13, 40.1920.65-40.ir.1.14, 40.1920.65-40.ir.1.15, 40.1920.65-40.ir.1.16, 40.1920.65-40.ir.1.17, 40.1920.65-40.ir.1.18, 40.1920.65-40.ir.1.19, 40.1920.65-40.ir.1.20, 40.1920.65-40.ir.1.21, 40.1920.65-40.ir.1.22, 40.1920.65-40.ir.1.23, 40.1920.65-40.ir.1.24
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $768$

Jacobian

Conductor: $2^{290}\cdot5^{112}$
Simple: no
Squarefree: no
Decomposition: $1^{29}\cdot2^{10}\cdot4^{4}$
Newforms: 40.2.d.a, 50.2.a.b$^{4}$, 64.2.a.a, 100.2.a.a$^{3}$, 160.2.d.a, 200.2.a.c$^{2}$, 200.2.a.e$^{2}$, 200.2.d.b, 200.2.d.d, 320.2.a.a, 320.2.a.b, 320.2.a.c, 320.2.a.d, 320.2.a.e, 320.2.a.f, 320.2.a.g, 400.2.a.a, 400.2.a.c, 400.2.a.e, 400.2.a.f, 800.2.d.a$^{2}$, 800.2.d.b, 800.2.d.c$^{2}$, 800.2.d.d, 800.2.d.e$^{2}$, 1600.2.a.b, 1600.2.a.ba, 1600.2.a.g, 1600.2.a.j, 1600.2.a.n, 1600.2.a.p, 1600.2.a.s, 1600.2.a.x

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=3,13,17,31,47,127,157,293$, and therefore no rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
40.96.1.a.1 $40$ $10$ $10$ $1$ $0$ $1^{28}\cdot2^{10}\cdot4^{4}$
40.480.31.a.1 $40$ $2$ $2$ $31$ $6$ $1^{14}\cdot2^{6}\cdot4^{2}$
40.480.31.g.2 $40$ $2$ $2$ $31$ $2$ $1^{14}\cdot2^{6}\cdot4^{2}$
40.480.31.ee.2 $40$ $2$ $2$ $31$ $6$ $1^{14}\cdot2^{6}\cdot4^{2}$
40.480.31.eg.1 $40$ $2$ $2$ $31$ $6$ $1^{14}\cdot2^{6}\cdot4^{2}$
40.480.33.ev.1 $40$ $2$ $2$ $33$ $8$ $2^{8}\cdot4^{4}$
40.480.33.ja.2 $40$ $2$ $2$ $33$ $4$ $1^{16}\cdot2^{4}\cdot4^{2}$
40.480.33.jc.2 $40$ $2$ $2$ $33$ $4$ $1^{16}\cdot2^{4}\cdot4^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.1920.129.b.1 $40$ $2$ $2$ $129$ $25$ $1^{26}\cdot2^{11}\cdot4^{4}$
40.1920.129.f.1 $40$ $2$ $2$ $129$ $30$ $1^{26}\cdot2^{11}\cdot4^{4}$
40.1920.129.ef.2 $40$ $2$ $2$ $129$ $27$ $1^{26}\cdot2^{11}\cdot4^{4}$
40.1920.129.ej.2 $40$ $2$ $2$ $129$ $24$ $1^{26}\cdot2^{11}\cdot4^{4}$
40.1920.129.ij.1 $40$ $2$ $2$ $129$ $30$ $1^{26}\cdot2^{11}\cdot4^{4}$
40.1920.129.in.1 $40$ $2$ $2$ $129$ $23$ $1^{26}\cdot2^{11}\cdot4^{4}$
40.1920.129.mn.2 $40$ $2$ $2$ $129$ $24$ $1^{26}\cdot2^{11}\cdot4^{4}$
40.1920.129.mr.2 $40$ $2$ $2$ $129$ $27$ $1^{26}\cdot2^{11}\cdot4^{4}$
40.1920.137.hh.1 $40$ $2$ $2$ $137$ $26$ $1^{28}\cdot2^{10}\cdot4^{6}$
40.1920.137.ho.1 $40$ $2$ $2$ $137$ $21$ $1^{28}\cdot2^{10}\cdot4^{6}$
40.1920.137.hr.2 $40$ $2$ $2$ $137$ $28$ $1^{28}\cdot2^{10}\cdot4^{6}$
40.1920.137.hx.1 $40$ $2$ $2$ $137$ $23$ $1^{28}\cdot2^{10}\cdot4^{6}$
40.1920.137.hz.1 $40$ $2$ $2$ $137$ $24$ $1^{24}\cdot2^{10}\cdot4^{5}\cdot8$
40.1920.137.ib.1 $40$ $2$ $2$ $137$ $28$ $1^{24}\cdot2^{10}\cdot4^{5}\cdot8$
40.1920.137.ig.1 $40$ $2$ $2$ $137$ $26$ $1^{24}\cdot2^{10}\cdot4^{5}\cdot8$
40.1920.137.ii.1 $40$ $2$ $2$ $137$ $30$ $1^{24}\cdot2^{10}\cdot4^{5}\cdot8$
40.2880.193.ta.2 $40$ $3$ $3$ $193$ $34$ $1^{54}\cdot2^{13}\cdot4^{12}$