Invariants
| Level: | $24$ | $\SL_2$-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 | $4^{2}$ | ||
| 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: | 24.48.1.471 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&18\\4&13\end{bmatrix}$, $\begin{bmatrix}9&8\\14&11\end{bmatrix}$, $\begin{bmatrix}15&19\\20&13\end{bmatrix}$ |
| 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} - 2 y^{2} - 5 y z - 2 z^{2} - w^{2} $ |
| $=$ | $6 x^{2} + 3 y^{2} + 9 y z + 3 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 28 x^{2} y^{2} + 12 x^{2} z^{2} + 100 y^{4} - 60 y^{2} z^{2} + 9 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^2}\cdot\frac{5701653342yz^{11}-12926154150yz^{9}w^{2}+10748959200yz^{7}w^{4}-3915702000yz^{5}w^{6}+576450000yz^{3}w^{8}-25200000yzw^{10}+2395519515z^{12}-4265004294z^{10}w^{2}+1991657160z^{8}w^{4}+307351800z^{6}w^{6}-381510000z^{4}w^{8}+59400000z^{2}w^{10}-1600000w^{12}}{93854376yz^{11}-15614100yz^{9}w^{2}-8442900yz^{7}w^{4}-283500yz^{5}w^{6}+162500yz^{3}w^{8}+25000yzw^{10}+39432420z^{12}+12576168z^{10}w^{2}-4758345z^{8}w^{4}-1748100z^{6}w^{6}-111250z^{4}w^{8}+12500z^{2}w^{10}-3125w^{12}}$ |
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.h.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.0.q.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.r.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.es.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.fa.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.bk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.bs.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.ti.1 | $24$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
| 24.192.9.hk.1 | $24$ | $4$ | $4$ | $9$ | $2$ | $1^{8}$ |
| 48.96.5.cm.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.cz.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.di.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.96.5.do.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 120.240.17.ms.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.288.17.oil.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 240.96.5.il.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.iv.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.jm.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.ka.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |