Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.19 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&0\\0&7\end{bmatrix}$, $\begin{bmatrix}13&20\\10&7\end{bmatrix}$, $\begin{bmatrix}17&8\\6&7\end{bmatrix}$, $\begin{bmatrix}21&19\\8&15\end{bmatrix}$ |
| 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: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 396x + 3024 $ |
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^4}\cdot\frac{72x^{2}y^{6}-3611952x^{2}y^{4}z^{2}+39665811456x^{2}y^{2}z^{4}-124233381946368x^{2}z^{6}-3528xy^{6}z+111227904xy^{4}z^{3}-1015833436416xy^{2}z^{5}+2853708674125824xz^{7}-y^{8}+114048y^{6}z^{2}-1863440640y^{4}z^{4}+10030572693504y^{2}z^{6}-16354899266015232z^{8}}{z^{2}(x^{2}y^{4}-101952x^{2}y^{2}z^{2}+887583744x^{2}z^{4}-72xy^{4}z+3089664xy^{2}z^{3}-20388298752xz^{5}+3024y^{4}z^{2}-47775744y^{2}z^{4}+116847525888z^{6})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.bq.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.by.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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.n.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.cm.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.dy.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ej.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.kh.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.kj.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ld.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.lj.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.72.5.kn.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.ef.1 | $24$ | $4$ | $4$ | $5$ | $3$ | $1^{4}$ |
| 120.48.1.bhf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bhj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bhv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bhz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.brz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bsd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bsp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bst.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.qj.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.oab.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.ewl.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bhd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bhh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bht.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bhx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.brx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bsb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bsn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bsr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.kh.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.bhd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bhh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bht.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bhx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.brx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bsb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bsn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bsr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.il.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bhf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bhj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bhv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bhz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.brz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bsd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bsp.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bst.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |