Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
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: | 8G1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.246 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&4\\0&17\end{bmatrix}$, $\begin{bmatrix}7&11\\2&23\end{bmatrix}$, $\begin{bmatrix}15&13\\22&21\end{bmatrix}$ |
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: | $2^{5}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 8 x^{2} + y^{2} + z^{2} - z w + w^{2} $ |
$=$ | $8 x y - z^{2} - 2 z w + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 8 x^{3} z + 8 x^{2} y^{2} - 4 x y^{2} z - 4 x z^{3} + 2 y^{4} + 2 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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
$\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{664y^{2}z^{10}-6104y^{2}z^{9}w+22104y^{2}z^{8}w^{2}-42816y^{2}z^{7}w^{3}+48576y^{2}z^{6}w^{4}-32832y^{2}z^{5}w^{5}+10944y^{2}z^{4}w^{6}+768y^{2}z^{3}w^{7}-1152y^{2}z^{2}w^{8}+640y^{2}zw^{9}-128y^{2}w^{10}+87z^{12}-204z^{11}w-1344z^{10}w^{2}+5368z^{9}w^{3}-4572z^{8}w^{4}-9312z^{7}w^{5}+25216z^{6}w^{6}-24384z^{5}w^{7}+12432z^{4}w^{8}-3520z^{3}w^{9}+384zw^{11}-64w^{12}}{(z^{2}+2zw-2w^{2})^{4}(8y^{2}z^{2}-8y^{2}zw+8y^{2}w^{2}+7z^{4}-20z^{3}w+24z^{2}w^{2}-8zw^{3}+4w^{4})}$ |
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.en.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.ee.1 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
24.144.9.eec.2 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
24.192.9.ql.2 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
48.96.5.hr.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.ht.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.jn.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.jp.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.96.5.rz.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.sb.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.tv.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.tx.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
120.240.17.fnu.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.cgwm.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
240.96.5.bzb.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzd.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzj.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzl.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqp.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqr.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqx.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqz.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |