Properties

Label 56.1344.93.ft.1
Level $56$
Index $1344$
Genus $93$
Analytic rank $12$
Cusps $40$
$\Q$-cusps $2$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $1344$ $\PSL_2$-index:$1344$
Genus: $93 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$
Cusps: $40$ (of which $2$ are rational) Cusp widths $14^{16}\cdot28^{8}\cdot56^{16}$ Cusp orbits $1^{2}\cdot2^{2}\cdot3^{2}\cdot4\cdot6^{2}\cdot12$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $12$
$\Q$-gonality: $14 \le \gamma \le 28$
$\overline{\Q}$-gonality: $14 \le \gamma \le 28$
Rational cusps: $2$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.1344.93.25

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}9&14\\0&51\end{bmatrix}$, $\begin{bmatrix}22&19\\47&34\end{bmatrix}$, $\begin{bmatrix}38&49\\53&2\end{bmatrix}$, $\begin{bmatrix}45&0\\10&39\end{bmatrix}$, $\begin{bmatrix}47&4\\10&37\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.2688.93-56.ft.1.1, 56.2688.93-56.ft.1.2, 56.2688.93-56.ft.1.3, 56.2688.93-56.ft.1.4, 56.2688.93-56.ft.1.5, 56.2688.93-56.ft.1.6, 56.2688.93-56.ft.1.7, 56.2688.93-56.ft.1.8, 56.2688.93-56.ft.1.9, 56.2688.93-56.ft.1.10, 56.2688.93-56.ft.1.11, 56.2688.93-56.ft.1.12, 56.2688.93-56.ft.1.13, 56.2688.93-56.ft.1.14, 56.2688.93-56.ft.1.15, 56.2688.93-56.ft.1.16
Cyclic 56-isogeny field degree: $2$
Cyclic 56-torsion field degree: $48$
Full 56-torsion field degree: $2304$

Jacobian

Conductor: $2^{375}\cdot7^{163}$
Simple: no
Squarefree: no
Decomposition: $1^{21}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$
Newforms: 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 56.2.b.a, 56.2.b.b, 98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 224.2.b.a, 224.2.b.b, 392.2.a.c, 392.2.a.f, 392.2.a.g, 392.2.b.e, 392.2.b.g, 448.2.a.a, 448.2.a.c, 448.2.a.d, 448.2.a.e, 448.2.a.g, 448.2.a.h, 1568.2.b.f, 1568.2.b.g, 3136.2.a.b, 3136.2.a.bc, 3136.2.a.bk, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.h, 3136.2.a.j, 3136.2.a.s, 3136.2.a.u

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal 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
$X_{\mathrm{sp}}^+(7)$ $7$ $48$ $48$ $0$ $0$ full Jacobian
8.48.0.m.2 $8$ $28$ $28$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.0.m.2 $8$ $28$ $28$ $0$ $0$ full Jacobian
56.672.45.cr.1 $56$ $2$ $2$ $45$ $12$ $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$
56.672.45.gq.2 $56$ $2$ $2$ $45$ $1$ $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$
56.672.45.gx.2 $56$ $2$ $2$ $45$ $1$ $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.2688.185.ra.2 $56$ $2$ $2$ $185$ $27$ $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.rd.1 $56$ $2$ $2$ $185$ $33$ $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.rg.2 $56$ $2$ $2$ $185$ $25$ $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.rj.1 $56$ $2$ $2$ $185$ $33$ $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.rn.1 $56$ $2$ $2$ $185$ $24$ $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.rp.1 $56$ $2$ $2$ $185$ $31$ $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.rt.1 $56$ $2$ $2$ $185$ $20$ $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.2688.185.rv.1 $56$ $2$ $2$ $185$ $35$ $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$
56.4032.277.ix.1 $56$ $3$ $3$ $277$ $41$ $1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$