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: | $4$ (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: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8C1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.70 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&4\\6&13\end{bmatrix}$, $\begin{bmatrix}11&10\\16&3\end{bmatrix}$, $\begin{bmatrix}17&9\\22&11\end{bmatrix}$, $\begin{bmatrix}19&20\\20&23\end{bmatrix}$ |
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: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x y + 2 x z - y w + z w $ |
$=$ | $8 x^{2} + 2 y^{2} + 2 y z + 2 z^{2} + 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 3 x^{3} z + 10 x^{2} y^{2} + 4 x^{2} z^{2} + 4 x y^{2} z + 3 x z^{3} + 10 y^{2} z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{192xz^{4}w-160xz^{2}w^{3}+72y^{2}z^{4}-24y^{2}z^{2}w^{2}-14y^{2}w^{4}+72yz^{5}-24yz^{3}w^{2}-46yzw^{4}+8z^{6}+156z^{4}w^{2}+6z^{2}w^{4}-27w^{6}}{64xz^{4}w-160xz^{2}w^{3}-8y^{2}z^{4}+72y^{2}z^{2}w^{2}-2y^{2}w^{4}-8yz^{5}+8yz^{3}w^{2}+110yzw^{4}-8z^{6}+52z^{4}w^{2}+18z^{2}w^{4}+27w^{6}}$ |
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 |
---|---|---|---|---|---|---|
8.12.0.q.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
12.12.0.o.1 | $12$ | $2$ | $2$ | $0$ | $0$ | 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.bp.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.48.1.ck.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.48.1.dh.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.48.1.dm.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.48.1.jp.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.48.1.jt.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.48.1.ke.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.48.1.ki.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.72.5.kf.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
24.96.5.dx.1 | $24$ | $4$ | $4$ | $5$ | $2$ | $1^{4}$ |
120.48.1.bfz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bgd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bgp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bgt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bqt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.bqx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.brj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.brn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.120.9.qb.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
120.144.9.nzt.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
120.240.17.ewd.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
168.48.1.bfx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bgb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bgn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bgr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bqr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.bqv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.brh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.brl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.13.jz.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
264.48.1.bfx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.bgb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.bgn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.bgr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.bqr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.bqv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.brh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.brl.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.288.21.id.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
312.48.1.bfz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.bgd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.bgp.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.bgt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.bqt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.bqx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.brj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.brn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |