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: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.18.1.2 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&17\\20&5\end{bmatrix}$, $\begin{bmatrix}5&23\\16&17\end{bmatrix}$, $\begin{bmatrix}11&16\\14&23\end{bmatrix}$, $\begin{bmatrix}15&19\\20&9\end{bmatrix}$, $\begin{bmatrix}23&15\\12&19\end{bmatrix}$ |
| 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.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 24x - 56 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(2:0:1)$ |
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.ef.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.eg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.el.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.em.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.et.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.ev.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.ew.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.ey.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.2.h.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.t.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.be.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.bg.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.dn.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.dp.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.dq.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.ds.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.dw.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.dy.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.ec.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.ee.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.ei.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.ek.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.eo.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.eq.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 72.54.3.j.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 72.162.11.a.1 | $72$ | $9$ | $9$ | $11$ | $?$ | not computed |
| 120.36.1.mj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.mk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.mp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.mq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.nh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.ni.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.nn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.no.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.2.ia.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ic.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ig.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ii.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.im.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.io.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.is.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.iu.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.iy.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ja.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.je.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jg.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jk.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jm.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jq.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.js.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.90.7.y.1 | $120$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 120.108.7.q.1 | $120$ | $6$ | $6$ | $7$ | $?$ | not computed |
| 120.180.13.bmm.1 | $120$ | $10$ | $10$ | $13$ | $?$ | not computed |
| 168.36.1.lx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.ly.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.md.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.me.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.mv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.mw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.nb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.nc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.2.hz.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ib.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.if.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ih.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.il.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.in.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ir.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.it.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ix.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.iz.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jd.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jf.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jj.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jl.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jp.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.jr.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.144.11.a.1 | $168$ | $8$ | $8$ | $11$ | $?$ | not computed |
| 264.36.1.lo.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.lp.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.lu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.lv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.mm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.mn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.ms.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.mt.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.2.ia.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ic.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ig.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ii.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.im.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.io.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.is.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.iu.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.iy.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ja.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.je.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jg.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jk.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jm.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.jq.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.js.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.216.17.a.1 | $264$ | $12$ | $12$ | $17$ | $?$ | not computed |
| 312.36.1.lx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.ly.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.md.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.me.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.mv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.mw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.nb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.nc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.2.ia.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ic.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ig.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ii.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.im.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.io.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.is.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.iu.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.iy.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ja.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.je.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jg.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jk.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jm.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.jq.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.js.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.252.19.bg.1 | $312$ | $14$ | $14$ | $19$ | $?$ | not computed |