Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.208 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&13\\16&15\end{bmatrix}$, $\begin{bmatrix}13&7\\10&17\end{bmatrix}$, $\begin{bmatrix}19&5\\10&7\end{bmatrix}$, $\begin{bmatrix}23&3\\6&5\end{bmatrix}$ |
| 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: | $2048$ |
Jacobian
| Conductor: | $2^{4}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 y^{2} - 4 z^{2} + 2 z w - w^{2} $ |
| $=$ | $12 x^{2} + z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 6 x^{2} z^{2} - 6 y^{2} z^{2} + 36 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{12}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(2z^{3}+w^{3})^{3}}{w^{3}z^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.1.i.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.18.0.a.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.n.1 | $24$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.72.3.o.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.dd.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.gw.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.gz.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.je.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.jg.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ku.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.kw.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 72.108.5.q.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.324.21.h.1 | $72$ | $9$ | $9$ | $21$ | $?$ | not computed |
| 120.72.3.eic.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eid.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eiq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eir.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ekg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ekh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eku.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ekv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.bqt.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bwn.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.dwk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dwl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dwy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dwz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dyo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dyp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dzc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dzd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.21.bbn.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.3.dwk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dwl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dwy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dwz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dyo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dyp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dzc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dzd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dwk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dwl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dwy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dwz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dyo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dyp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dzc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dzd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |