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Properties

Label 56.48.1.q.2
Level $56$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $32$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.48.1.267

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}9&0\\18&15\end{bmatrix}$, $\begin{bmatrix}21&32\\50&43\end{bmatrix}$, $\begin{bmatrix}29&28\\42&55\end{bmatrix}$, $\begin{bmatrix}37&14\\48&23\end{bmatrix}$, $\begin{bmatrix}39&18\\12&41\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.96.1-56.q.2.1, 56.96.1-56.q.2.2, 56.96.1-56.q.2.3, 56.96.1-56.q.2.4, 56.96.1-56.q.2.5, 56.96.1-56.q.2.6, 56.96.1-56.q.2.7, 56.96.1-56.q.2.8, 56.96.1-56.q.2.9, 56.96.1-56.q.2.10, 56.96.1-56.q.2.11, 56.96.1-56.q.2.12, 56.96.1-56.q.2.13, 56.96.1-56.q.2.14, 56.96.1-56.q.2.15, 56.96.1-56.q.2.16, 112.96.1-56.q.2.1, 112.96.1-56.q.2.2, 112.96.1-56.q.2.3, 112.96.1-56.q.2.4, 112.96.1-56.q.2.5, 112.96.1-56.q.2.6, 112.96.1-56.q.2.7, 112.96.1-56.q.2.8, 168.96.1-56.q.2.1, 168.96.1-56.q.2.2, 168.96.1-56.q.2.3, 168.96.1-56.q.2.4, 168.96.1-56.q.2.5, 168.96.1-56.q.2.6, 168.96.1-56.q.2.7, 168.96.1-56.q.2.8, 168.96.1-56.q.2.9, 168.96.1-56.q.2.10, 168.96.1-56.q.2.11, 168.96.1-56.q.2.12, 168.96.1-56.q.2.13, 168.96.1-56.q.2.14, 168.96.1-56.q.2.15, 168.96.1-56.q.2.16, 280.96.1-56.q.2.1, 280.96.1-56.q.2.2, 280.96.1-56.q.2.3, 280.96.1-56.q.2.4, 280.96.1-56.q.2.5, 280.96.1-56.q.2.6, 280.96.1-56.q.2.7, 280.96.1-56.q.2.8, 280.96.1-56.q.2.9, 280.96.1-56.q.2.10, 280.96.1-56.q.2.11, 280.96.1-56.q.2.12, 280.96.1-56.q.2.13, 280.96.1-56.q.2.14, 280.96.1-56.q.2.15, 280.96.1-56.q.2.16
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $64512$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 7 x^{2} + 7 x y - 2 z^{2} $
$=$ $7 x^{2} - 7 x y + 14 y^{2} - 2 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 49 x^{4} - 7 x^{2} y^{2} + 21 x^{2} z^{2} + 2 z^{4} $
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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.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{903168y^{2}z^{10}-677376y^{2}z^{8}w^{2}+16128y^{2}z^{6}w^{4}+4032y^{2}z^{4}w^{6}-10584y^{2}z^{2}w^{8}+882y^{2}w^{10}-131072z^{12}+196608z^{10}w^{2}-65280z^{8}w^{4}+4096z^{6}w^{6}-768z^{4}w^{8}-240z^{2}w^{10}+31w^{12}}{w^{4}z^{4}(56y^{2}z^{2}+14y^{2}w^{2}+w^{4})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.1.c.1 $8$ $2$ $2$ $1$ $0$ dimension zero
56.24.0.h.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.24.0.i.2 $56$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.96.1.c.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.d.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.f.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.k.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.q.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.v.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.y.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.z.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.br.1 $56$ $8$ $8$ $25$ $1$ $1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.de.2 $56$ $21$ $21$ $73$ $9$ $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.de.2 $56$ $28$ $28$ $97$ $10$ $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
112.96.5.bd.2 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.be.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.bi.2 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.bl.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.bo.2 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.br.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.bu.2 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.bv.1 $112$ $2$ $2$ $5$ $?$ not computed
168.96.1.j.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.k.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bc.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bh.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.cb.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.cg.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.cv.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.cw.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.144.9.dn.2 $168$ $3$ $3$ $9$ $?$ not computed
168.192.9.ca.2 $168$ $4$ $4$ $9$ $?$ not computed
280.96.1.j.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.k.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.bc.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.bh.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.cb.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.cg.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.cv.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.cw.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.240.17.bh.2 $280$ $5$ $5$ $17$ $?$ not computed
280.288.17.ck.1 $280$ $6$ $6$ $17$ $?$ not computed