Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $18$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $1 = 1 + \frac{ 18 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot12$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $2$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 12C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.18.1.1 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&13\\10&3\end{bmatrix}$, $\begin{bmatrix}7&4\\8&7\end{bmatrix}$, $\begin{bmatrix}11&3\\12&7\end{bmatrix}$, $\begin{bmatrix}11&16\\22&19\end{bmatrix}$, $\begin{bmatrix}13&16\\4&17\end{bmatrix}$ |
| 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: | $4096$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 24x + 56 $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 18 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^6}\cdot\frac{18x^{2}y^{4}-3240x^{2}y^{2}z^{2}+122472x^{2}z^{4}+72xy^{4}z-3240xy^{2}z^{3}-244944xz^{5}+y^{6}-198y^{4}z^{2}+26892y^{2}z^{4}-822312z^{6}}{z^{4}(6x^{2}-12xz+y^{2}-48z^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 6.9.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.36.1.ei.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.36.1.ej.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.36.1.eo.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.36.1.ep.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.36.1.eq.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.36.1.es.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.36.1.ez.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.36.1.fb.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.36.2.g.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.s.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.bn.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.bp.1 | $24$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 24.36.2.dk.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.dm.1 | $24$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 24.36.2.dt.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.dv.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.dz.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.eb.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.ef.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.eh.1 | $24$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 24.36.2.el.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.en.1 | $24$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 24.36.2.er.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.et.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 72.54.3.i.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.162.11.b.1 | $72$ | $9$ | $9$ | $11$ | $?$ | not computed |
| 120.36.1.mm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.mn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.ms.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.mt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.nk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.nl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.nq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.nr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.2.id.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.if.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ij.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.il.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ip.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ir.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.iv.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ix.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jb.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jd.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jh.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jj.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jn.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jp.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jt.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jv.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.90.7.z.1 | $120$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 120.108.7.r.1 | $120$ | $6$ | $6$ | $7$ | $?$ | not computed |
| 120.180.13.bmn.1 | $120$ | $10$ | $10$ | $13$ | $?$ | not computed |
| 168.36.1.ma.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.mb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.mg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.mh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.my.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.mz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.ne.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.nf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.2.ic.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ie.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ii.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ik.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.io.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.iq.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.iu.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.iw.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ja.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jc.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jg.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ji.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jm.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jo.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.js.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ju.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.144.11.b.1 | $168$ | $8$ | $8$ | $11$ | $?$ | not computed |
| 264.36.1.lr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.ls.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.lx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.ly.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.mp.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.mq.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.mv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.mw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.2.id.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.if.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ij.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.il.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ip.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ir.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.iv.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ix.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jb.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jd.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jh.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jj.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jn.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jp.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jt.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jv.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.216.17.b.1 | $264$ | $12$ | $12$ | $17$ | $?$ | not computed |
| 312.36.1.ma.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.mb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.mg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.mh.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.my.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.mz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.ne.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.nf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.2.id.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.if.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ij.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.il.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ip.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ir.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.iv.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ix.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jb.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jd.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jh.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jj.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jn.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jp.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jt.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jv.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.252.19.bh.1 | $312$ | $14$ | $14$ | $19$ | $?$ | not computed |