Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
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: | $1$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.249 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&4\\0&19\end{bmatrix}$, $\begin{bmatrix}21&1\\22&11\end{bmatrix}$, $\begin{bmatrix}21&23\\22&17\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + x y + 3 x z + y^{2} - 3 y z + 2 z^{2} + w^{2} $ |
$=$ | $x^{2} + 4 x y + 2 x z + y^{2} - 2 y z - 4 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 20 x^{4} + 24 x^{3} y + 48 x^{2} y^{2} + 10 x^{2} z^{2} - 36 x y^{3} + 9 y^{4} + 12 y^{2} z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{1728xz^{9}w^{2}+2304xz^{7}w^{4}+224xz^{5}w^{6}-320xz^{3}w^{8}+12xzw^{10}-1728yz^{9}w^{2}-2304yz^{7}w^{4}-224yz^{5}w^{6}+320yz^{3}w^{8}-12yzw^{10}-1728z^{12}+5040z^{8}w^{4}+3200z^{6}w^{6}+92z^{4}w^{8}-96z^{2}w^{10}+w^{12}}{z^{8}(4xzw^{2}-4yzw^{2}-16z^{4}+w^{4})}$ |
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 |
---|---|---|---|---|---|---|
8.24.0.bk.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.em.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.ef.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
24.144.9.eeg.2 | $24$ | $3$ | $3$ | $9$ | $3$ | $1^{4}\cdot2^{2}$ |
24.192.9.qn.2 | $24$ | $4$ | $4$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
48.96.5.hv.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.hx.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.jr.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.96.5.jt.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.96.5.sd.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.sf.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.tz.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.96.5.ub.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
120.240.17.fny.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.cgws.2 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
240.96.5.bzf.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzh.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzn.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzp.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqt.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqv.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.crb.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.crd.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |