Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| 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: | 24.36.1.215 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&22\\10&13\end{bmatrix}$, $\begin{bmatrix}9&16\\10&19\end{bmatrix}$, $\begin{bmatrix}13&6\\10&17\end{bmatrix}$, $\begin{bmatrix}17&7\\16&13\end{bmatrix}$ |
| 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: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 4 x^{2} - 3 y w $ |
| $=$ | $4 y^{2} - 2 y w - 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 6 x^{2} z^{2} - 2 y^{2} z^{2} + 4 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{3}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{3}{4}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{32yz^{8}+96yz^{6}w^{2}+216yz^{4}w^{4}+108yz^{2}w^{6}+64z^{8}w+48z^{6}w^{3}+54z^{2}w^{7}+27w^{9}}{w^{3}(24yz^{4}w-8yz^{2}w^{3}+8z^{6}-4z^{2}w^{4}+w^{6})}$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.1.g.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.18.0.d.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.m.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.bv.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.cu.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.fy.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ga.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.md.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.me.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.mk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ml.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 72.108.5.s.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.324.21.m.1 | $72$ | $9$ | $9$ | $21$ | $?$ | not computed |
| 120.72.3.elg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.elh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eln.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.elo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.enk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.enl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.enr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ens.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.bqy.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bws.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.dzo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dzp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dzv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dzw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ebs.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ebt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ebz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eca.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.21.bbs.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.3.dzo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dzp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dzv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dzw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ebs.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ebt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ebz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eca.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dzo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dzp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dzv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dzw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ebs.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ebt.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ebz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eca.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |