Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-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 | $2^{2}\cdot4$ | ||
| 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: | 24.96.1.903 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&16\\20&21\end{bmatrix}$, $\begin{bmatrix}5&6\\16&1\end{bmatrix}$, $\begin{bmatrix}11&23\\4&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1087283 |
| Contains $-I$: | no $\quad$ (see 24.48.1.fa.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
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 $ | $=$ | $ x^{2} - x z + 6 y^{2} + z^{2} $ |
| $=$ | $7 x^{2} - 4 x z + 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 20 x^{2} y^{2} + 3 x^{2} z^{2} + 196 y^{4} + 84 y^{2} z^{2} + 9 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{30195296640xz^{11}-63407183616xz^{9}w^{2}+28300176576xz^{7}w^{4}+385850304xz^{5}w^{6}-642565224xz^{3}w^{8}+121010400xzw^{10}-9922751424z^{12}+51334523712z^{10}w^{2}-59671379376z^{8}w^{4}+25404182496z^{6}w^{6}-4834279044z^{4}w^{8}+230351940z^{2}w^{10}+2100875w^{12}}{31065120xz^{11}+7764120xz^{9}w^{2}-12674340xz^{7}w^{4}+1382976xz^{5}w^{6}+941192xz^{3}w^{8}-67228xzw^{10}-10208592z^{12}-24629832z^{10}w^{2}+5387823z^{8}w^{4}+2916186z^{6}w^{6}-929187z^{4}w^{8}+4802z^{2}w^{10}+16807w^{12}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.fa.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+20X^{2}Y^{2}+196Y^{4}+3X^{2}Z^{2}+84Y^{2}Z^{2}+9Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.n.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.48.0-12.e.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.e.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bd.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bd.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-8.n.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.288.9-24.big.1.1 | $24$ | $3$ | $3$ | $9$ | $2$ | $1^{8}$ |
| 24.384.9-24.la.1.1 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
| 48.192.5-48.gp.1.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.5-48.gp.2.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.192.5-48.gq.1.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.192.5-48.gq.2.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 120.480.17-120.vq.1.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 240.192.5-240.ul.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.ul.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.um.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5-240.um.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |