Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.68 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&8\\22&3\end{bmatrix}$, $\begin{bmatrix}7&5\\8&17\end{bmatrix}$, $\begin{bmatrix}11&8\\4&23\end{bmatrix}$, $\begin{bmatrix}17&1\\8&23\end{bmatrix}$ |
| 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: | $3072$ |
Jacobian
| Conductor: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 99x - 378 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{84x^{2}y^{6}+10593x^{2}y^{4}z^{2}+812275128x^{2}y^{2}z^{4}+681664256343x^{2}z^{6}-1902xy^{6}z+2377728xy^{4}z^{3}+11624899599xy^{2}z^{5}+7829107075062xz^{7}-y^{8}-2096y^{6}z^{2}+52427520y^{4}z^{4}+79493405508y^{2}z^{6}+22434728690583z^{8}}{60x^{2}y^{6}+340767x^{2}y^{4}z^{2}+17496x^{2}y^{2}z^{4}+59049x^{2}z^{6}+1710xy^{6}z+3930768xy^{4}z^{3}-98415xy^{2}z^{5}-354294xz^{7}+y^{8}+27216y^{6}z^{2}+11337408y^{4}z^{4}-1180980y^{2}z^{6}-3720087z^{8}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.v.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.bs.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.bz.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 |
|---|---|---|---|---|---|---|
| 24.48.1.bl.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.db.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fr.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.la.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lm.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.lr.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.md.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.mo.1 | $24$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
| 24.96.5.fa.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 48.48.2.cq.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.cs.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.do.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.dq.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.em.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.eo.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.eu.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.ew.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 120.48.1.bjk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bjo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bkq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bku.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bue.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bui.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bvk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bvo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.sk.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.odi.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.fay.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bji.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bjm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bko.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bks.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.buc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bug.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bvi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bvm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.mi.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 240.48.2.dq.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ds.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.dy.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ea.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ew.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ey.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.fe.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.fg.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.48.1.bji.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bjm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bko.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bks.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.buc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bug.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bvi.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bvm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.km.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bjk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bjo.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bkq.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bku.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bue.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bui.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bvk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bvo.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |