Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $64$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $12\cdot24$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $8$ 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: | 24D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.68 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&6\\12&13\end{bmatrix}$, $\begin{bmatrix}5&12\\12&1\end{bmatrix}$, $\begin{bmatrix}17&3\\6&17\end{bmatrix}$, $\begin{bmatrix}23&1\\14&17\end{bmatrix}$, $\begin{bmatrix}23&4\\4&7\end{bmatrix}$ |
| 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: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x $ |
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 |
|---|
| $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{960x^{2}y^{8}z^{2}-1056x^{2}y^{4}z^{6}+28x^{2}z^{10}-384xy^{10}z+2048xy^{6}z^{5}-296xy^{2}z^{9}+64y^{12}-1232y^{8}z^{4}+268y^{4}z^{8}+z^{12}}{z^{8}(xz-y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.0.k.1 | $12$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.72.1.dk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.dn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.do.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.dr.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.ea.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.ed.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.ee.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.eh.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.k.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.y.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.bk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.bs.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.72.5.ds.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.dv.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.dw.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.dz.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.ie.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.ih.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.ii.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.il.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.72.5.iu.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.ix.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.72.5.iy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.jb.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.108.7.ge.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.gn.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.1.ul.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.un.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.up.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ur.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.vr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.vt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.vv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.vx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.byb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bzx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bzz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cab.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cad.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.can.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cap.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.car.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cat.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.180.13.bwj.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.chb.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.in.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.ip.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.ir.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.it.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.jt.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.jv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.jx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.jz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.biv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bix.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.biz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bkr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bkt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bkv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bkx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.blh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.blj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bll.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bln.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.23.bp.1 | $168$ | $8$ | $8$ | $23$ | $?$ | not computed |
| 264.72.1.ij.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.il.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.in.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.ip.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.jp.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.jr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.jt.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.jv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.5.biv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bix.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.biz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjp.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bkr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bkt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bkv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bkx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.blh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.blj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bll.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bln.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.1.in.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.ip.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.ir.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.it.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.jt.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.jv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.jx.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.jz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.5.biv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bix.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.biz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bkr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bkt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bkv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bkx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.blh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.blj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bll.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bln.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |