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: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.349 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}5&52\\12&11\end{bmatrix}$, $\begin{bmatrix}7&40\\32&15\end{bmatrix}$, $\begin{bmatrix}21&16\\36&23\end{bmatrix}$, $\begin{bmatrix}26&7\\45&18\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.96.1-56.fv.1.1, 56.96.1-56.fv.1.2, 168.96.1-56.fv.1.1, 168.96.1-56.fv.1.2, 280.96.1-56.fv.1.1, 280.96.1-56.fv.1.2 |
| 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 $ | $=$ | $ 4 y^{2} + 6 y z + y w + 4 z^{2} - z w $ |
| $=$ | $7 x^{2} + y^{2} - 2 y z + 2 y w + z^{2} - 2 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 23 x^{4} + 48 x^{3} z - 7 x^{2} y^{2} + 54 x^{2} z^{2} + 14 x y^{2} z + 48 x z^{3} - 7 y^{2} z^{2} + 23 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot7^4}\cdot\frac{10842531840yz^{11}-59633925120yz^{10}w+136796610048yz^{9}w^{2}-168330306816yz^{8}w^{3}+119479906560yz^{7}w^{4}-50075716992yz^{6}w^{5}+9217227712yz^{5}w^{6}+7545132896yz^{4}w^{7}-11126860024yz^{3}w^{8}+6876615508yz^{2}w^{9}-1297450714yzw^{10}-146882449yw^{11}+5300793344z^{12}-26383494144z^{11}w+56742583296z^{10}w^{2}-72855790336z^{9}w^{3}+74315598336z^{8}w^{4}-76510842240z^{7}w^{5}+63947273600z^{6}w^{6}-28883837920z^{5}w^{7}+1338585136z^{4}w^{8}+3748034500z^{3}w^{9}-758903572z^{2}w^{10}+146882449zw^{11}-268435456w^{12}}{w^{4}(384yz^{7}-1344yz^{6}w+1120yz^{5}w^{2}+560yz^{4}w^{3}-952yz^{3}w^{4}+196yz^{2}w^{5}+34yzw^{6}+yw^{7}+1280z^{8}-4928z^{7}w+6720z^{6}w^{2}-3696z^{5}w^{3}+560z^{4}w^{4}+84z^{3}w^{5}-20z^{2}w^{6}-zw^{7})}$ |
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.x.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 28.24.0.i.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.cb.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.da.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.dj.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.1.be.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.24.1.bm.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.nh.1 | $56$ | $8$ | $8$ | $25$ | $7$ | $1^{20}\cdot2^{2}$ |
| 56.1008.73.bnv.1 | $56$ | $21$ | $21$ | $73$ | $21$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.1344.97.bnb.1 | $56$ | $28$ | $28$ | $97$ | $28$ | $1^{36}\cdot2^{28}\cdot4$ |
| 168.144.9.ewp.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.192.9.bog.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.240.17.yv.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 280.288.17.dfp.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |