Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $8$ 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: | 12T1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.42 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&8\\20&15\end{bmatrix}$, $\begin{bmatrix}3&14\\16&9\end{bmatrix}$, $\begin{bmatrix}5&3\\12&5\end{bmatrix}$, $\begin{bmatrix}15&22\\10&9\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y + y^{2} + z^{2} $ |
| $=$ | $6 x^{2} + 16 x y - 8 y^{2} - 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} - 6 x^{2} y^{2} + 2 x^{2} z^{2} - 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot3^6}\cdot\frac{(3456z^{6}+w^{6})^{3}}{w^{6}z^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.0.c.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.e.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.ei.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 |
|---|---|---|---|---|---|---|
| 24.144.5.x.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bn.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.db.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.dd.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.fo.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.ft.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.gy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.hi.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.9.i.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.q.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.ca.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.144.5.esa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.esc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eso.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.esq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eue.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eug.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eus.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.euu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cea.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cec.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ceo.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ceq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cge.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cgg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cgs.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cgu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cea.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cec.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ceo.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ceq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cge.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cgg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cgs.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cgu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cea.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cec.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ceo.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ceq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cge.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cgg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cgs.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cgu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |