Properties

Label 56.48.0-56.n.1.5
Level $56$
Index $48$
Genus $0$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $8$
Index: $48$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (none of which are rational) Cusp widths $2^{4}\cdot8^{2}$ Cusp orbits $2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1 \le \gamma \le 2$
$\overline{\Q}$-gonality: $1$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.48.0.872

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}3&20\\36&11\end{bmatrix}$, $\begin{bmatrix}27&46\\0&9\end{bmatrix}$, $\begin{bmatrix}31&41\\48&43\end{bmatrix}$
Contains $-I$: no $\quad$ (see 56.24.0.n.1 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $64512$

Models

Smooth plane model Smooth plane model

$ 0 $ $=$ $ x^{2} - 14 y^{2} - 112 z^{2} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.24.0-8.d.1.4 $8$ $2$ $2$ $0$ $0$
56.24.0-8.d.1.2 $56$ $2$ $2$ $0$ $0$
56.24.0-56.y.1.1 $56$ $2$ $2$ $0$ $0$
56.24.0-56.y.1.12 $56$ $2$ $2$ $0$ $0$
56.24.0-56.ba.1.1 $56$ $2$ $2$ $0$ $0$
56.24.0-56.ba.1.15 $56$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
56.384.11-56.br.1.2 $56$ $8$ $8$ $11$
56.1008.34-56.cj.1.2 $56$ $21$ $21$ $34$
56.1344.45-56.cl.1.1 $56$ $28$ $28$ $45$
168.144.4-168.eu.1.1 $168$ $3$ $3$ $4$
168.192.3-168.fu.1.1 $168$ $4$ $4$ $3$
280.240.8-280.bm.1.1 $280$ $5$ $5$ $8$
280.288.7-280.dg.1.17 $280$ $6$ $6$ $7$
280.480.15-280.eu.1.17 $280$ $10$ $10$ $15$