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Properties

Label 56.48.1.x.1
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: $64$
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: 56.48.1.283

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}1&4\\54&31\end{bmatrix}$, $\begin{bmatrix}25&28\\12&51\end{bmatrix}$, $\begin{bmatrix}27&24\\0&51\end{bmatrix}$, $\begin{bmatrix}39&44\\24&21\end{bmatrix}$, $\begin{bmatrix}41&8\\16&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.96.1-56.x.1.1, 56.96.1-56.x.1.2, 56.96.1-56.x.1.3, 56.96.1-56.x.1.4, 56.96.1-56.x.1.5, 56.96.1-56.x.1.6, 56.96.1-56.x.1.7, 56.96.1-56.x.1.8, 56.96.1-56.x.1.9, 56.96.1-56.x.1.10, 56.96.1-56.x.1.11, 56.96.1-56.x.1.12, 56.96.1-56.x.1.13, 56.96.1-56.x.1.14, 56.96.1-56.x.1.15, 56.96.1-56.x.1.16, 168.96.1-56.x.1.1, 168.96.1-56.x.1.2, 168.96.1-56.x.1.3, 168.96.1-56.x.1.4, 168.96.1-56.x.1.5, 168.96.1-56.x.1.6, 168.96.1-56.x.1.7, 168.96.1-56.x.1.8, 168.96.1-56.x.1.9, 168.96.1-56.x.1.10, 168.96.1-56.x.1.11, 168.96.1-56.x.1.12, 168.96.1-56.x.1.13, 168.96.1-56.x.1.14, 168.96.1-56.x.1.15, 168.96.1-56.x.1.16, 280.96.1-56.x.1.1, 280.96.1-56.x.1.2, 280.96.1-56.x.1.3, 280.96.1-56.x.1.4, 280.96.1-56.x.1.5, 280.96.1-56.x.1.6, 280.96.1-56.x.1.7, 280.96.1-56.x.1.8, 280.96.1-56.x.1.9, 280.96.1-56.x.1.10, 280.96.1-56.x.1.11, 280.96.1-56.x.1.12, 280.96.1-56.x.1.13, 280.96.1-56.x.1.14, 280.96.1-56.x.1.15, 280.96.1-56.x.1.16
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $64512$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

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

$ 0 $ $=$ $ 2 x^{4} + 14 x^{2} y^{2} - 21 x^{2} z^{2} + 49 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{7}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{7}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{14112y^{2}z^{10}-21168y^{2}z^{8}w^{2}+1008y^{2}z^{6}w^{4}+504y^{2}z^{4}w^{6}-2646y^{2}z^{2}w^{8}+441y^{2}w^{10}+2048z^{12}-6144z^{10}w^{2}+4080z^{8}w^{4}-512z^{6}w^{6}+192z^{4}w^{8}+120z^{2}w^{10}-31w^{12}}{w^{4}z^{4}(14y^{2}z^{2}+7y^{2}w^{2}-w^{4})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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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.d.1 $8$ $2$ $2$ $1$ $0$ dimension zero
56.24.0.h.2 $56$ $2$ $2$ $0$ $0$ 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.e.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.p.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.ba.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bg.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bi.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bo.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bq.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.br.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.cj.1 $56$ $8$ $8$ $25$ $3$ $1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.ed.1 $56$ $21$ $21$ $73$ $9$ $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.ed.2 $56$ $28$ $28$ $97$ $12$ $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
168.96.1.de.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.dg.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.du.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ee.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ek.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.eu.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.fi.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.fk.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.144.9.ep.1 $168$ $3$ $3$ $9$ $?$ not computed
168.192.9.cv.1 $168$ $4$ $4$ $9$ $?$ not computed
280.96.1.de.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.dg.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.du.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ee.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ek.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.eu.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.fi.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.fk.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.240.17.bv.2 $280$ $5$ $5$ $17$ $?$ not computed
280.288.17.df.1 $280$ $6$ $6$ $17$ $?$ not computed