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 | $4^{2}$ | ||
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: | 56.48.1.415 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}12&13\\15&20\end{bmatrix}$, $\begin{bmatrix}18&19\\31&26\end{bmatrix}$, $\begin{bmatrix}21&4\\44&39\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 56-isogeny field degree: | $32$ |
Cyclic 56-torsion field degree: | $768$ |
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 $ | $=$ | $ 14 x^{2} + 5 y^{2} + y z - z^{2} + w^{2} $ |
$=$ | $13 y^{2} + 4 y z - 4 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 60 x^{2} y^{2} + 56 x^{2} z^{2} + 676 y^{4} + 1092 y^{2} z^{2} + 441 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{7}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{7^2}\cdot\frac{27376831710270yz^{11}-83079200935458yz^{9}w^{2}+91076046633936yz^{7}w^{4}-42541036206032yz^{5}w^{6}+7678402759664yz^{3}w^{8}-424165123200yzw^{10}-11591440549731z^{12}+39009755381436z^{10}w^{2}-49851443173896z^{8}w^{4}+29663145453032z^{6}w^{6}-8018894557856z^{4}w^{8}+852744634560z^{2}w^{10}-23762752000w^{12}}{82771977960yz^{11}+14717999660yz^{9}w^{2}-14805268660yz^{7}w^{4}-4553598868yz^{5}w^{6}+150459348yz^{3}w^{8}+160398576yzw^{10}-35045927588z^{12}+4503531004z^{10}w^{2}+9321347763z^{8}w^{4}+676562432z^{6}w^{6}-593923798z^{4}w^{8}-163483164z^{2}w^{10}-30074733w^{12}}$ |
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.h.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.0.q.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.s.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.em.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.eu.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.1.bm.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.1.bu.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.hm.1 | $56$ | $8$ | $8$ | $25$ | $6$ | $1^{20}\cdot2^{2}$ |
56.1008.73.ts.1 | $56$ | $21$ | $21$ | $73$ | $38$ | $1^{16}\cdot2^{26}\cdot4$ |
56.1344.97.tk.1 | $56$ | $28$ | $28$ | $97$ | $44$ | $1^{36}\cdot2^{28}\cdot4$ |
112.96.5.co.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.db.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.dk.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.dq.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.9.cfy.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.192.9.xd.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
280.240.17.ma.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
280.288.17.bix.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |