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Properties

Label 40.1440.101.cp.1
Level $40$
Index $1440$
Genus $101$
Analytic rank $38$
Cusps $36$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $40$ Newform level: $1600$
Index: $1440$ $\PSL_2$-index:$1440$
Genus: $101 = 1 + \frac{ 1440 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$
Cusps: $36$ (none of which are rational) Cusp widths $40^{36}$ Cusp orbits $4^{5}\cdot8^{2}$
Elliptic points: $8$ of order $2$ and $0$ of order $3$
Analytic rank: $38$
$\Q$-gonality: $26 \le \gamma \le 32$
$\overline{\Q}$-gonality: $26 \le \gamma \le 32$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.1440.101.79

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&9\\0&23\end{bmatrix}$, $\begin{bmatrix}13&12\\34&7\end{bmatrix}$, $\begin{bmatrix}23&0\\34&17\end{bmatrix}$, $\begin{bmatrix}39&24\\12&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 40-isogeny field degree: $8$
Cyclic 40-torsion field degree: $128$
Full 40-torsion field degree: $512$

Jacobian

Conductor: $2^{504}\cdot5^{175}$
Simple: no
Squarefree: no
Decomposition: $1^{83}\cdot2^{9}$
Newforms: 20.2.a.a$^{2}$, 32.2.a.a, 40.2.a.a$^{2}$, 50.2.a.a, 50.2.a.b$^{2}$, 64.2.a.a, 80.2.a.a$^{2}$, 80.2.a.b, 100.2.a.a$^{2}$, 160.2.a.a, 160.2.a.b, 160.2.a.c, 200.2.a.a, 200.2.a.c$^{2}$, 200.2.a.e, 320.2.a.a, 320.2.a.b$^{2}$, 320.2.a.c$^{2}$, 320.2.a.d, 320.2.a.e$^{2}$, 320.2.a.f$^{2}$, 320.2.a.g, 400.2.a.a$^{2}$, 400.2.a.b, 400.2.a.c$^{2}$, 400.2.a.d, 400.2.a.e$^{3}$, 400.2.a.f, 400.2.a.g, 400.2.a.h, 800.2.a.a$^{2}$, 800.2.a.b, 800.2.a.c, 800.2.a.d$^{2}$, 800.2.a.g, 800.2.a.h, 800.2.a.i$^{2}$, 800.2.a.j, 800.2.a.l, 800.2.a.m, 800.2.a.n, 1600.2.a.a$^{2}$, 1600.2.a.b, 1600.2.a.ba, 1600.2.a.bd, 1600.2.a.c, 1600.2.a.d$^{2}$, 1600.2.a.e, 1600.2.a.f$^{2}$, 1600.2.a.g, 1600.2.a.h, 1600.2.a.i$^{2}$, 1600.2.a.j, 1600.2.a.k, 1600.2.a.l, 1600.2.a.m, 1600.2.a.o$^{2}$, 1600.2.a.p, 1600.2.a.q, 1600.2.a.r, 1600.2.a.s, 1600.2.a.t, 1600.2.a.u$^{2}$, 1600.2.a.v$^{2}$, 1600.2.a.w, 1600.2.a.x, 1600.2.a.y, 1600.2.a.z

Rational points

This modular curve has no $\Q_p$ points for $p=13$, 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
40.720.47.bbn.1 $40$ $2$ $2$ $47$ $21$ $1^{40}\cdot2^{7}$
40.720.49.evr.1 $40$ $2$ $2$ $49$ $18$ $1^{40}\cdot2^{6}$
40.720.49.evv.1 $40$ $2$ $2$ $49$ $21$ $1^{42}\cdot2^{5}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.2880.205.lv.2 $40$ $2$ $2$ $205$ $63$ $1^{80}\cdot2^{12}$
40.2880.205.bnb.1 $40$ $2$ $2$ $205$ $65$ $1^{80}\cdot2^{12}$
40.2880.205.bpt.1 $40$ $2$ $2$ $205$ $72$ $1^{80}\cdot2^{12}$
40.2880.205.bqd.1 $40$ $2$ $2$ $205$ $74$ $1^{80}\cdot2^{12}$
40.2880.205.cgt.1 $40$ $2$ $2$ $205$ $71$ $1^{80}\cdot2^{12}$
40.2880.205.chk.1 $40$ $2$ $2$ $205$ $76$ $1^{80}\cdot2^{12}$
40.2880.205.chp.1 $40$ $2$ $2$ $205$ $69$ $1^{80}\cdot2^{12}$
40.2880.205.cii.1 $40$ $2$ $2$ $205$ $74$ $1^{80}\cdot2^{12}$