Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| 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: | 24.36.1.154 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&10\\20&7\end{bmatrix}$, $\begin{bmatrix}17&13\\16&7\end{bmatrix}$, $\begin{bmatrix}21&11\\10&3\end{bmatrix}$, $\begin{bmatrix}23&23\\2&7\end{bmatrix}$ |
| 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: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y w $ |
| $=$ | $x^{2} + 4 y^{2} - y w - 3 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 2 x^{2} z^{2} - 3 y^{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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{6yz^{8}+12yz^{6}w^{2}+18yz^{4}w^{4}+6yz^{2}w^{6}+12z^{8}w+6z^{6}w^{3}+3z^{2}w^{7}+w^{9}}{w^{3}(54yz^{4}w-12yz^{2}w^{3}+27z^{6}-6z^{2}w^{4}+w^{6})}$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.0.a.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.o.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.i.1 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.72.3.j.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.cb.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.fl.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.fm.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.rf.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.rh.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.rt.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.rv.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.5.bf.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.324.21.bb.1 | $72$ | $9$ | $9$ | $21$ | $?$ | not computed |
| 120.72.3.eqh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eqj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eqv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eqx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.esl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.esn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.esz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.etb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.brn.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bxh.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.eep.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eer.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.efd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eff.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.egt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.egv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ehh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ehj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.21.bch.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.3.eep.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eer.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.efd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eff.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.egt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.egv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ehh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ehj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eep.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eer.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.efd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eff.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.egt.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.egv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ehh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ehj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |