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: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.403 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}15&8\\12&25\end{bmatrix}$, $\begin{bmatrix}40&7\\53&12\end{bmatrix}$, $\begin{bmatrix}46&23\\13&6\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 112.96.1-56.ge.1.1, 112.96.1-56.ge.1.2, 112.96.1-56.ge.1.3, 112.96.1-56.ge.1.4, 112.96.1-56.ge.1.5, 112.96.1-56.ge.1.6 |
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 $ | $=$ | $ 5 y^{2} - 8 y z - 8 z^{2} + 2 w^{2} $ |
$=$ | $28 x^{2} - y^{2} + 3 y z + 3 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 18 x^{2} y^{2} - 56 x^{2} z^{2} + 25 y^{4} - 210 y^{2} z^{2} + 441 z^{4} $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{7}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{4256915649714yz^{11}-286318169550yz^{9}w^{2}-77973915600yz^{7}w^{4}+753914000yz^{5}w^{6}+384650000yz^{3}w^{8}+8400000yzw^{10}+2965643167599z^{12}-768293819496z^{10}w^{2}-41427910440z^{8}w^{4}+10387206200z^{6}w^{6}+527240000z^{4}w^{8}-21000000z^{2}w^{10}-1600000w^{12}}{630654170328yz^{11}-452345278700yz^{9}w^{2}+121282433300yz^{7}w^{4}-14624319500yz^{5}w^{6}+729487500yz^{3}w^{8}-9450000yzw^{10}+439354543348z^{12}-399410892692z^{10}w^{2}+141182749645z^{8}w^{4}-24034764600z^{6}w^{6}+1942298750z^{4}w^{8}-59062500z^{2}w^{10}+253125w^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.24.1.y.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.0.ch.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.cm.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dg.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.0.dw.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
56.24.1.z.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
56.24.1.bg.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.nw.1 | $56$ | $8$ | $8$ | $25$ | $11$ | $1^{20}\cdot2^{2}$ |
56.1008.73.bpk.1 | $56$ | $21$ | $21$ | $73$ | $32$ | $1^{16}\cdot2^{26}\cdot4$ |
56.1344.97.boq.1 | $56$ | $28$ | $28$ | $97$ | $43$ | $1^{36}\cdot2^{28}\cdot4$ |
112.96.3.nf.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.ng.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.nh.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
112.96.3.ni.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.144.9.fbg.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.192.9.bqt.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
280.240.17.bba.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
280.288.17.djq.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |