Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $6^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ 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: | 6E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.127 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&7\\2&23\end{bmatrix}$, $\begin{bmatrix}9&7\\20&3\end{bmatrix}$, $\begin{bmatrix}15&10\\14&3\end{bmatrix}$, $\begin{bmatrix}23&7\\22&1\end{bmatrix}$ |
| 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.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 2 x y + 2 x z + 2 z^{2} $ |
| $=$ | $x^{2} + x y + x z + 8 y^{2} + z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 7 x^{4} + 10 x^{3} z - 2 x^{2} y^{2} + 9 x^{2} z^{2} + 4 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^3\cdot3\,\frac{3759912xz^{8}-13019940xz^{6}w^{2}+14946750xz^{4}w^{4}-2487877xz^{2}w^{6}-812567xw^{8}-84589056y^{2}z^{7}+230078016y^{2}z^{5}w^{2}-116988480y^{2}z^{3}w^{4}+25711280y^{2}zw^{6}+2770848yz^{8}-36602496yz^{6}w^{2}+17389512yz^{4}w^{4}-6676152yz^{2}w^{6}+1647086yw^{8}-1181952z^{9}+18197136z^{7}w^{2}-29281896z^{5}w^{4}+14375228z^{3}w^{6}-3213910zw^{8}}{52221xz^{8}+405477xz^{6}w^{2}+398804xz^{4}w^{4}+29792xz^{2}w^{6}-4116xw^{8}-1174848y^{2}z^{7}-2763264y^{2}z^{5}w^{2}-1386112y^{2}z^{3}w^{4}+43904y^{2}zw^{6}+38484yz^{8}+686742yz^{6}w^{2}+310464yz^{4}w^{4}+32928yz^{2}w^{6}-16416z^{9}+242154z^{7}w^{2}+704480z^{5}w^{4}+217168z^{3}w^{6}-5488zw^{8}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.18.0.e.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.r.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.18.0.d.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.a.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.cg.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ci.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ct.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.cu.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.dz.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ea.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.eg.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ei.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.7.f.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.j.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.3.is.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.iu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ja.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.jc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.me.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.mg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.mm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.mo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.u.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bo.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.ii.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ik.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.iq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.is.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.lm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.lo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.lu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.lw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.qp.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.3.ii.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ik.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.iq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.is.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.lm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.lo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.lu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.lw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ii.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ik.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.iq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.is.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.lm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.lo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.lu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.lw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |