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Properties

Label 40.240.15.z.1
Level $40$
Index $240$
Genus $15$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $40$ Newform level: $200$
Index: $240$ $\PSL_2$-index:$240$
Genus: $15 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $10^{4}\cdot20^{6}\cdot40^{2}$ Cusp orbits $2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4 \le \gamma \le 8$
$\overline{\Q}$-gonality: $4 \le \gamma \le 8$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 40O15
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.240.15.1

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&4\\18&9\end{bmatrix}$, $\begin{bmatrix}5&4\\34&25\end{bmatrix}$, $\begin{bmatrix}5&16\\34&9\end{bmatrix}$, $\begin{bmatrix}13&32\\18&1\end{bmatrix}$, $\begin{bmatrix}15&16\\36&25\end{bmatrix}$, $\begin{bmatrix}15&24\\26&1\end{bmatrix}$, $\begin{bmatrix}39&16\\24&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.480.15-40.z.1.1, 40.480.15-40.z.1.2, 40.480.15-40.z.1.3, 40.480.15-40.z.1.4, 40.480.15-40.z.1.5, 40.480.15-40.z.1.6, 40.480.15-40.z.1.7, 40.480.15-40.z.1.8, 40.480.15-40.z.1.9, 40.480.15-40.z.1.10, 40.480.15-40.z.1.11, 40.480.15-40.z.1.12, 40.480.15-40.z.1.13, 40.480.15-40.z.1.14, 40.480.15-40.z.1.15, 40.480.15-40.z.1.16, 40.480.15-40.z.1.17, 40.480.15-40.z.1.18, 40.480.15-40.z.1.19, 40.480.15-40.z.1.20, 40.480.15-40.z.1.21, 40.480.15-40.z.1.22, 40.480.15-40.z.1.23, 40.480.15-40.z.1.24, 40.480.15-40.z.1.25, 40.480.15-40.z.1.26, 40.480.15-40.z.1.27, 40.480.15-40.z.1.28, 40.480.15-40.z.1.29, 40.480.15-40.z.1.30, 40.480.15-40.z.1.31, 40.480.15-40.z.1.32, 40.480.15-40.z.1.33, 40.480.15-40.z.1.34, 40.480.15-40.z.1.35, 40.480.15-40.z.1.36, 40.480.15-40.z.1.37, 40.480.15-40.z.1.38, 40.480.15-40.z.1.39, 40.480.15-40.z.1.40, 40.480.15-40.z.1.41, 40.480.15-40.z.1.42, 40.480.15-40.z.1.43, 40.480.15-40.z.1.44, 40.480.15-40.z.1.45, 40.480.15-40.z.1.46, 40.480.15-40.z.1.47, 40.480.15-40.z.1.48, 40.480.15-40.z.1.49, 40.480.15-40.z.1.50, 40.480.15-40.z.1.51, 40.480.15-40.z.1.52, 40.480.15-40.z.1.53, 40.480.15-40.z.1.54, 40.480.15-40.z.1.55, 40.480.15-40.z.1.56, 40.480.15-40.z.1.57, 40.480.15-40.z.1.58, 40.480.15-40.z.1.59, 40.480.15-40.z.1.60, 40.480.15-40.z.1.61, 40.480.15-40.z.1.62, 40.480.15-40.z.1.63, 40.480.15-40.z.1.64, 120.480.15-40.z.1.1, 120.480.15-40.z.1.2, 120.480.15-40.z.1.3, 120.480.15-40.z.1.4, 120.480.15-40.z.1.5, 120.480.15-40.z.1.6, 120.480.15-40.z.1.7, 120.480.15-40.z.1.8, 120.480.15-40.z.1.9, 120.480.15-40.z.1.10, 120.480.15-40.z.1.11, 120.480.15-40.z.1.12, 120.480.15-40.z.1.13, 120.480.15-40.z.1.14, 120.480.15-40.z.1.15, 120.480.15-40.z.1.16, 120.480.15-40.z.1.17, 120.480.15-40.z.1.18, 120.480.15-40.z.1.19, 120.480.15-40.z.1.20, 120.480.15-40.z.1.21, 120.480.15-40.z.1.22, 120.480.15-40.z.1.23, 120.480.15-40.z.1.24, 120.480.15-40.z.1.25, 120.480.15-40.z.1.26, 120.480.15-40.z.1.27, 120.480.15-40.z.1.28, 120.480.15-40.z.1.29, 120.480.15-40.z.1.30, 120.480.15-40.z.1.31, 120.480.15-40.z.1.32, 120.480.15-40.z.1.33, 120.480.15-40.z.1.34, 120.480.15-40.z.1.35, 120.480.15-40.z.1.36, 120.480.15-40.z.1.37, 120.480.15-40.z.1.38, 120.480.15-40.z.1.39, 120.480.15-40.z.1.40, 120.480.15-40.z.1.41, 120.480.15-40.z.1.42, 120.480.15-40.z.1.43, 120.480.15-40.z.1.44, 120.480.15-40.z.1.45, 120.480.15-40.z.1.46, 120.480.15-40.z.1.47, 120.480.15-40.z.1.48, 120.480.15-40.z.1.49, 120.480.15-40.z.1.50, 120.480.15-40.z.1.51, 120.480.15-40.z.1.52, 120.480.15-40.z.1.53, 120.480.15-40.z.1.54, 120.480.15-40.z.1.55, 120.480.15-40.z.1.56, 120.480.15-40.z.1.57, 120.480.15-40.z.1.58, 120.480.15-40.z.1.59, 120.480.15-40.z.1.60, 120.480.15-40.z.1.61, 120.480.15-40.z.1.62, 120.480.15-40.z.1.63, 120.480.15-40.z.1.64, 280.480.15-40.z.1.1, 280.480.15-40.z.1.2, 280.480.15-40.z.1.3, 280.480.15-40.z.1.4, 280.480.15-40.z.1.5, 280.480.15-40.z.1.6, 280.480.15-40.z.1.7, 280.480.15-40.z.1.8, 280.480.15-40.z.1.9, 280.480.15-40.z.1.10, 280.480.15-40.z.1.11, 280.480.15-40.z.1.12, 280.480.15-40.z.1.13, 280.480.15-40.z.1.14, 280.480.15-40.z.1.15, 280.480.15-40.z.1.16, 280.480.15-40.z.1.17, 280.480.15-40.z.1.18, 280.480.15-40.z.1.19, 280.480.15-40.z.1.20, 280.480.15-40.z.1.21, 280.480.15-40.z.1.22, 280.480.15-40.z.1.23, 280.480.15-40.z.1.24, 280.480.15-40.z.1.25, 280.480.15-40.z.1.26, 280.480.15-40.z.1.27, 280.480.15-40.z.1.28, 280.480.15-40.z.1.29, 280.480.15-40.z.1.30, 280.480.15-40.z.1.31, 280.480.15-40.z.1.32, 280.480.15-40.z.1.33, 280.480.15-40.z.1.34, 280.480.15-40.z.1.35, 280.480.15-40.z.1.36, 280.480.15-40.z.1.37, 280.480.15-40.z.1.38, 280.480.15-40.z.1.39, 280.480.15-40.z.1.40, 280.480.15-40.z.1.41, 280.480.15-40.z.1.42, 280.480.15-40.z.1.43, 280.480.15-40.z.1.44, 280.480.15-40.z.1.45, 280.480.15-40.z.1.46, 280.480.15-40.z.1.47, 280.480.15-40.z.1.48, 280.480.15-40.z.1.49, 280.480.15-40.z.1.50, 280.480.15-40.z.1.51, 280.480.15-40.z.1.52, 280.480.15-40.z.1.53, 280.480.15-40.z.1.54, 280.480.15-40.z.1.55, 280.480.15-40.z.1.56, 280.480.15-40.z.1.57, 280.480.15-40.z.1.58, 280.480.15-40.z.1.59, 280.480.15-40.z.1.60, 280.480.15-40.z.1.61, 280.480.15-40.z.1.62, 280.480.15-40.z.1.63, 280.480.15-40.z.1.64
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $3072$

Jacobian

Conductor: $2^{37}\cdot5^{30}$
Simple: no
Squarefree: no
Decomposition: $1^{7}\cdot2^{2}\cdot4$
Newforms: 50.2.a.b$^{3}$, 100.2.a.a$^{2}$, 200.2.a.c, 200.2.a.e, 200.2.d.a, 200.2.d.c, 200.2.d.f

Rational points

This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{ns}}^+(5)$ $5$ $24$ $24$ $0$ $0$ full Jacobian
8.24.0.e.2 $8$ $10$ $10$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.e.2 $8$ $10$ $10$ $0$ $0$ full Jacobian
20.120.7.b.1 $20$ $2$ $2$ $7$ $0$ $2^{2}\cdot4$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.480.29.dz.1 $40$ $2$ $2$ $29$ $1$ $1^{6}\cdot2^{2}\cdot4$
40.480.29.ed.1 $40$ $2$ $2$ $29$ $6$ $1^{6}\cdot2^{2}\cdot4$
40.480.29.eh.1 $40$ $2$ $2$ $29$ $3$ $1^{6}\cdot2^{2}\cdot4$
40.480.29.el.1 $40$ $2$ $2$ $29$ $0$ $1^{6}\cdot2^{2}\cdot4$
40.480.29.ep.1 $40$ $2$ $2$ $29$ $1$ $1^{6}\cdot2^{2}\cdot4$
40.480.29.et.1 $40$ $2$ $2$ $29$ $4$ $1^{6}\cdot2^{2}\cdot4$
40.480.29.ex.1 $40$ $2$ $2$ $29$ $3$ $1^{6}\cdot2^{2}\cdot4$
40.480.29.fb.1 $40$ $2$ $2$ $29$ $4$ $1^{6}\cdot2^{2}\cdot4$
40.480.31.h.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.l.1 $40$ $2$ $2$ $31$ $2$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.t.1 $40$ $2$ $2$ $31$ $6$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.x.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.bh.2 $40$ $2$ $2$ $31$ $1$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.bj.1 $40$ $2$ $2$ $31$ $1$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.bp.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.br.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.bx.1 $40$ $2$ $2$ $31$ $2$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.bz.2 $40$ $2$ $2$ $31$ $2$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.cf.2 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.ch.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.cn.2 $40$ $2$ $2$ $31$ $2$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.cp.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.cv.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.cx.2 $40$ $2$ $2$ $31$ $6$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.dd.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.df.2 $40$ $2$ $2$ $31$ $6$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.dl.2 $40$ $2$ $2$ $31$ $2$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.dn.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.dt.2 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.dv.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.eb.1 $40$ $2$ $2$ $31$ $2$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.ed.2 $40$ $2$ $2$ $31$ $2$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.ej.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.el.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.er.1 $40$ $2$ $2$ $31$ $1$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.et.1 $40$ $2$ $2$ $31$ $1$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.fb.1 $40$ $2$ $2$ $31$ $6$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.ff.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.fn.1 $40$ $2$ $2$ $31$ $4$ $1^{8}\cdot2^{2}\cdot4$
40.480.31.fr.1 $40$ $2$ $2$ $31$ $2$ $1^{8}\cdot2^{2}\cdot4$
40.480.33.fh.1 $40$ $2$ $2$ $33$ $5$ $1^{6}\cdot2^{4}\cdot4$
40.480.33.fp.1 $40$ $2$ $2$ $33$ $3$ $1^{6}\cdot2^{4}\cdot4$
40.480.33.gg.1 $40$ $2$ $2$ $33$ $6$ $1^{6}\cdot2^{4}\cdot4$
40.480.33.gk.1 $40$ $2$ $2$ $33$ $4$ $1^{6}\cdot2^{4}\cdot4$
40.480.33.ix.1 $40$ $2$ $2$ $33$ $3$ $1^{6}\cdot2^{4}\cdot4$
40.480.33.iz.1 $40$ $2$ $2$ $33$ $3$ $1^{6}\cdot2^{4}\cdot4$
40.480.33.jf.1 $40$ $2$ $2$ $33$ $2$ $1^{6}\cdot2^{4}\cdot4$
40.480.33.jh.1 $40$ $2$ $2$ $33$ $2$ $1^{6}\cdot2^{4}\cdot4$
40.480.33.md.1 $40$ $2$ $2$ $33$ $4$ $1^{4}\cdot2^{5}\cdot4$
40.480.33.mf.2 $40$ $2$ $2$ $33$ $6$ $1^{4}\cdot2^{5}\cdot4$
40.480.33.ml.2 $40$ $2$ $2$ $33$ $4$ $1^{4}\cdot2^{5}\cdot4$
40.480.33.mn.1 $40$ $2$ $2$ $33$ $6$ $1^{4}\cdot2^{5}\cdot4$
40.480.33.mt.2 $40$ $2$ $2$ $33$ $4$ $1^{4}\cdot2^{5}\cdot4$
40.480.33.mv.1 $40$ $2$ $2$ $33$ $4$ $1^{4}\cdot2^{5}\cdot4$
40.480.33.nb.1 $40$ $2$ $2$ $33$ $4$ $1^{4}\cdot2^{5}\cdot4$
40.480.33.nd.1 $40$ $2$ $2$ $33$ $4$ $1^{4}\cdot2^{5}\cdot4$
40.720.43.fl.1 $40$ $3$ $3$ $43$ $1$ $1^{12}\cdot2^{2}\cdot4^{3}$