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Properties

Label 56.96.0-8.c.1.3
Level $56$
Index $96$
Genus $0$
Analytic rank $0$
Cusps $10$
$\Q$-cusps $4$

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Invariants

Level: $56$ $\SL_2$-level: $8$
Index: $96$ $\PSL_2$-index:$48$
Genus: $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $4$ are rational) Cusp widths $4^{8}\cdot8^{2}$ Cusp orbits $1^{4}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8N0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.96.0.232

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}5&40\\8&41\end{bmatrix}$, $\begin{bmatrix}7&20\\44&15\end{bmatrix}$, $\begin{bmatrix}19&16\\40&17\end{bmatrix}$, $\begin{bmatrix}27&36\\32&23\end{bmatrix}$, $\begin{bmatrix}55&20\\48&25\end{bmatrix}$
Contains $-I$: no $\quad$ (see 8.48.0.c.1 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $32256$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 6 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 48 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{x^{48}(x^{8}-4x^{7}y+4x^{6}y^{2}+28x^{5}y^{3}+6x^{4}y^{4}-28x^{3}y^{5}+4x^{2}y^{6}+4xy^{7}+y^{8})^{3}(x^{8}+4x^{7}y+4x^{6}y^{2}-28x^{5}y^{3}+6x^{4}y^{4}+28x^{3}y^{5}+4x^{2}y^{6}-4xy^{7}+y^{8})^{3}}{y^{4}x^{52}(x-y)^{4}(x+y)^{4}(x^{2}+y^{2})^{8}(x^{2}-2xy-y^{2})^{4}(x^{2}+2xy-y^{2})^{4}}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
56.48.0-4.b.1.4 $56$ $2$ $2$ $0$ $0$
56.48.0-4.b.1.6 $56$ $2$ $2$ $0$ $0$
56.48.0-8.e.2.6 $56$ $2$ $2$ $0$ $0$
56.48.0-8.e.2.8 $56$ $2$ $2$ $0$ $0$
56.48.0-8.e.2.9 $56$ $2$ $2$ $0$ $0$
56.48.0-8.e.2.11 $56$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
56.192.1-8.f.1.3 $56$ $2$ $2$ $1$
56.192.1-8.f.2.2 $56$ $2$ $2$ $1$
56.192.1-8.f.2.3 $56$ $2$ $2$ $1$
56.192.1-8.g.1.2 $56$ $2$ $2$ $1$
56.192.1-8.g.2.1 $56$ $2$ $2$ $1$
56.192.1-8.g.2.6 $56$ $2$ $2$ $1$
56.192.3-8.i.1.2 $56$ $2$ $2$ $3$
56.192.3-8.j.1.4 $56$ $2$ $2$ $3$
112.192.2-16.a.1.5 $112$ $2$ $2$ $2$
112.192.2-16.a.1.6 $112$ $2$ $2$ $2$
112.192.2-16.b.1.10 $112$ $2$ $2$ $2$
112.192.2-16.b.1.12 $112$ $2$ $2$ $2$
112.192.2-16.c.1.6 $112$ $2$ $2$ $2$
112.192.2-16.c.1.14 $112$ $2$ $2$ $2$
112.192.2-16.d.1.3 $112$ $2$ $2$ $2$
112.192.2-16.d.1.7 $112$ $2$ $2$ $2$
168.192.1-24.w.1.11 $168$ $2$ $2$ $1$
168.192.1-24.w.2.3 $168$ $2$ $2$ $1$
168.192.1-24.w.2.5 $168$ $2$ $2$ $1$
168.192.1-24.x.1.9 $168$ $2$ $2$ $1$
168.192.1-24.x.2.2 $168$ $2$ $2$ $1$
168.192.1-24.x.2.8 $168$ $2$ $2$ $1$
168.192.3-24.z.1.2 $168$ $2$ $2$ $3$
168.192.3-24.ba.1.6 $168$ $2$ $2$ $3$
168.288.8-24.l.1.4 $168$ $3$ $3$ $8$
168.384.7-24.i.1.3 $168$ $4$ $4$ $7$
280.192.1-40.w.1.11 $280$ $2$ $2$ $1$
280.192.1-40.w.2.1 $280$ $2$ $2$ $1$
280.192.1-40.w.2.7 $280$ $2$ $2$ $1$
280.192.1-40.x.1.9 $280$ $2$ $2$ $1$
280.192.1-40.x.2.4 $280$ $2$ $2$ $1$
280.192.1-40.x.2.6 $280$ $2$ $2$ $1$
280.192.3-40.be.1.2 $280$ $2$ $2$ $3$
280.192.3-40.bf.1.7 $280$ $2$ $2$ $3$
280.480.16-40.f.1.4 $280$ $5$ $5$ $16$
56.192.1-56.w.1.11 $56$ $2$ $2$ $1$
56.192.1-56.w.2.2 $56$ $2$ $2$ $1$
56.192.1-56.w.2.8 $56$ $2$ $2$ $1$
56.192.1-56.x.1.11 $56$ $2$ $2$ $1$
56.192.1-56.x.2.3 $56$ $2$ $2$ $1$
56.192.1-56.x.2.5 $56$ $2$ $2$ $1$
56.192.3-56.w.1.6 $56$ $2$ $2$ $3$
56.192.3-56.x.1.8 $56$ $2$ $2$ $3$
56.768.23-56.i.1.20 $56$ $8$ $8$ $23$
56.2016.70-56.l.1.23 $56$ $21$ $21$ $70$
56.2688.93-56.l.1.25 $56$ $28$ $28$ $93$
112.192.2-112.a.1.17 $112$ $2$ $2$ $2$
112.192.2-112.a.1.26 $112$ $2$ $2$ $2$
112.192.2-112.b.1.17 $112$ $2$ $2$ $2$
112.192.2-112.b.1.27 $112$ $2$ $2$ $2$
112.192.2-112.c.1.17 $112$ $2$ $2$ $2$
112.192.2-112.c.1.27 $112$ $2$ $2$ $2$
112.192.2-112.d.1.17 $112$ $2$ $2$ $2$
112.192.2-112.d.1.26 $112$ $2$ $2$ $2$
168.192.1-168.cy.1.23 $168$ $2$ $2$ $1$
168.192.1-168.cy.2.1 $168$ $2$ $2$ $1$
168.192.1-168.cy.2.15 $168$ $2$ $2$ $1$
168.192.1-168.cz.1.18 $168$ $2$ $2$ $1$
168.192.1-168.cz.2.8 $168$ $2$ $2$ $1$
168.192.1-168.cz.2.10 $168$ $2$ $2$ $1$
168.192.3-168.co.1.2 $168$ $2$ $2$ $3$
168.192.3-168.cp.1.6 $168$ $2$ $2$ $3$
280.192.1-280.cy.1.18 $280$ $2$ $2$ $1$
280.192.1-280.cy.2.2 $280$ $2$ $2$ $1$
280.192.1-280.cy.2.15 $280$ $2$ $2$ $1$
280.192.1-280.cz.1.18 $280$ $2$ $2$ $1$
280.192.1-280.cz.2.7 $280$ $2$ $2$ $1$
280.192.1-280.cz.2.10 $280$ $2$ $2$ $1$
280.192.3-280.cw.1.6 $280$ $2$ $2$ $3$
280.192.3-280.cx.1.7 $280$ $2$ $2$ $3$