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: | 24.48.1.252 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&5\\22&21\end{bmatrix}$, $\begin{bmatrix}19&2\\16&13\end{bmatrix}$, $\begin{bmatrix}19&10\\0&17\end{bmatrix}$ |
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: | $2^{5}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - x y + x z + y^{2} + y z + 2 z^{2} $ |
$=$ | $3 x y - 5 x z - 5 y z + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 116 x^{4} - 168 x^{3} y + 96 x^{2} y^{2} + 17 x^{2} z^{2} - 36 x y^{3} - 18 x y z^{2} + 9 y^{4} + \cdots + 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{55296xz^{9}w^{2}+36864xz^{7}w^{4}+1792xz^{5}w^{6}-1280xz^{3}w^{8}+24xzw^{10}+55296yz^{9}w^{2}+36864yz^{7}w^{4}+1792yz^{5}w^{6}-1280yz^{3}w^{8}+24yzw^{10}+110592z^{12}-80640z^{8}w^{4}-25600z^{6}w^{6}-368z^{4}w^{8}+192z^{2}w^{10}-w^{12}}{z^{8}(8xzw^{2}+8yzw^{2}+64z^{4}-w^{4})}$ |
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.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.en.1 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.1 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
24.192.9.ql.1 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
48.96.5.hr.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.ht.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.jn.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.jp.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.96.5.rz.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.sb.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.tv.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.tx.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
120.240.17.fnu.2 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.cgwm.2 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
240.96.5.bzb.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzd.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzj.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.bzl.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqp.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqr.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqx.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqz.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |