Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $12^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $4$ 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: | 12G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.124 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}2&23\\7&2\end{bmatrix}$, $\begin{bmatrix}5&15\\12&7\end{bmatrix}$, $\begin{bmatrix}18&17\\13&18\end{bmatrix}$, $\begin{bmatrix}20&1\\11&20\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $48$ |
| Cyclic 24-torsion field degree: | $384$ |
| Full 24-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x y - z^{2} $ |
| $=$ | $72 x^{2} - 24 x z + 2 y^{2} - 4 y z + 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 6 x^{3} z - 2 x^{2} y^{2} + x^{2} z^{2} - 2 x z^{3} + 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{3}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^7\cdot3^3\,\frac{720xz^{5}+4656xz^{3}w^{2}-80y^{2}z^{4}-540y^{2}z^{2}w^{2}-27y^{2}w^{4}+40yz^{5}-108yzw^{4}+52z^{6}-554z^{4}w^{2}}{1440xz^{5}+768xz^{3}w^{2}+72xzw^{4}-160y^{2}z^{4}-40y^{2}z^{2}w^{2}-2y^{2}w^{4}+80yz^{5}+32yz^{3}w^{2}+4yzw^{4}+104z^{6}+100z^{4}w^{2}+18z^{2}w^{4}+w^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.12.0.q.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.6.0.h.1 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.3.v.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.48.3.x.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.bb.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.48.3.bd.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.bw.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.48.3.by.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.cf.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.48.3.ch.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.bai.1 | $24$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 24.96.5.jb.1 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 72.72.3.cs.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.216.15.zn.1 | $72$ | $9$ | $9$ | $15$ | $?$ | not computed |
| 120.48.3.ej.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.el.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.ep.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.er.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.ex.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.ez.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.fd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.ff.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.120.9.bxf.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.bgzp.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.lqr.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.3.dp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.dr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.dv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.dx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.ed.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.ef.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.ej.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.el.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.15.eh.1 | $168$ | $8$ | $8$ | $15$ | $?$ | not computed |
| 264.48.3.dp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.dr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.dv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.dx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.ed.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.ef.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.ej.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.el.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.288.23.eh.1 | $264$ | $12$ | $12$ | $23$ | $?$ | not computed |
| 312.48.3.dp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.dr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.dv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.dx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.ed.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.ef.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.ej.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.el.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |