Properties

Label 56.24.1.b.1
Level $56$
Index $24$
Genus $1$
Analytic rank $1$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $3136$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $4^{2}\cdot8^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8B1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.24.1.4

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}13&34\\46&15\end{bmatrix}$, $\begin{bmatrix}31&8\\52&39\end{bmatrix}$, $\begin{bmatrix}35&10\\4&33\end{bmatrix}$, $\begin{bmatrix}43&12\\42&11\end{bmatrix}$, $\begin{bmatrix}53&20\\20&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.48.1-56.b.1.1, 56.48.1-56.b.1.2, 56.48.1-56.b.1.3, 56.48.1-56.b.1.4, 56.48.1-56.b.1.5, 56.48.1-56.b.1.6, 56.48.1-56.b.1.7, 56.48.1-56.b.1.8, 168.48.1-56.b.1.1, 168.48.1-56.b.1.2, 168.48.1-56.b.1.3, 168.48.1-56.b.1.4, 168.48.1-56.b.1.5, 168.48.1-56.b.1.6, 168.48.1-56.b.1.7, 168.48.1-56.b.1.8, 280.48.1-56.b.1.1, 280.48.1-56.b.1.2, 280.48.1-56.b.1.3, 280.48.1-56.b.1.4, 280.48.1-56.b.1.5, 280.48.1-56.b.1.6, 280.48.1-56.b.1.7, 280.48.1-56.b.1.8
Cyclic 56-isogeny field degree: $32$
Cyclic 56-torsion field degree: $768$
Full 56-torsion field degree: $129024$

Jacobian

Conductor: $2^{6}\cdot7^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 3136.2.a.m

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 49x $
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Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{7^4}\cdot\frac{7203x^{2}y^{4}z^{2}-49xy^{6}z+17294403xy^{2}z^{5}+y^{8}+13841287201z^{8}}{z^{2}y^{4}x^{2}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
4.12.0.a.1 $4$ $2$ $2$ $0$ $0$ full Jacobian
56.12.0.bx.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.12.1.c.1 $56$ $2$ $2$ $1$ $1$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.48.1.a.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.d.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.n.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.z.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.bk.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.bn.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.bp.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.br.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.192.13.f.1 $56$ $8$ $8$ $13$ $5$ $1^{8}\cdot2^{2}$
56.504.37.f.1 $56$ $21$ $21$ $37$ $13$ $1^{4}\cdot2^{14}\cdot4$
56.672.49.f.1 $56$ $28$ $28$ $49$ $17$ $1^{12}\cdot2^{16}\cdot4$
168.48.1.bd.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bf.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bp.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.br.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.en.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.ep.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.er.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.et.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.5.b.1 $168$ $3$ $3$ $5$ $?$ not computed
168.96.5.b.1 $168$ $4$ $4$ $5$ $?$ not computed
280.48.1.bd.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.bf.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.bp.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.br.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.ej.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.el.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.en.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.ep.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.120.9.b.1 $280$ $5$ $5$ $9$ $?$ not computed
280.144.9.b.1 $280$ $6$ $6$ $9$ $?$ not computed
280.240.17.fp.1 $280$ $10$ $10$ $17$ $?$ not computed