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 | $4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.316 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&43\\46&51\end{bmatrix}$, $\begin{bmatrix}25&30\\42&19\end{bmatrix}$, $\begin{bmatrix}45&53\\32&15\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 112.96.1-56.fk.1.1, 112.96.1-56.fk.1.2, 112.96.1-56.fk.1.3, 112.96.1-56.fk.1.4 |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
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 $ | $=$ | $ 14 x^{2} + 2 y^{2} - y z + z^{2} $ |
$=$ | $9 y^{2} - 8 y z + 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} - 20 x^{2} y^{2} + 21 x^{2} z^{2} + 81 y^{4} - 126 y^{2} z^{2} + 49 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{7}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot3^3\,\frac{204577493120yz^{11}-507798361728yz^{9}w^{2}-26773032384yz^{7}w^{4}+21059958528yz^{5}w^{6}+3130016904yz^{3}w^{8}+248005800yzw^{10}+319862218816z^{12}+326120338880z^{10}w^{2}-117033998256z^{8}w^{4}-35840514528z^{6}w^{6}-1416409092z^{4}w^{8}-35823060z^{2}w^{10}-7381125w^{12}}{51144373280yz^{11}+125905270680yz^{9}w^{2}+56675653020yz^{7}w^{4}+8208542916yz^{5}w^{6}+660981384yz^{3}w^{8}+39680928yzw^{10}+79965554704z^{12}+10989492248z^{10}w^{2}-25725752199z^{8}w^{4}-8604561798z^{6}w^{6}-988721559z^{4}w^{8}-47396664z^{2}w^{10}+944784w^{12}}$ |
Modular covers
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.q.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
28.24.0.h.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.ca.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dd.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dk.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.1.bc.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.1.bh.1 | $56$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
56.384.25.mw.1 | $56$ | $8$ | $8$ | $25$ | $9$ | $1^{20}\cdot2^{2}$ |
56.1008.73.bnk.1 | $56$ | $21$ | $21$ | $73$ | $19$ | $1^{16}\cdot2^{26}\cdot4$ |
56.1344.97.bmq.1 | $56$ | $28$ | $28$ | $97$ | $28$ | $1^{36}\cdot2^{28}\cdot4$ |
168.144.9.ewe.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.192.9.bnv.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
280.240.17.yk.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
280.288.17.dfe.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |