Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $32$ | ||
| 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: | yes $\quad(D =$ $-8$) |
Other labels
| Cummins and Pauli (CP) label: | 24D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.67 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&8\\8&7\end{bmatrix}$, $\begin{bmatrix}5&11\\2&7\end{bmatrix}$, $\begin{bmatrix}7&1\\14&7\end{bmatrix}$, $\begin{bmatrix}15&2\\16&21\end{bmatrix}$, $\begin{bmatrix}15&10\\16&21\end{bmatrix}$ |
| 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 4x $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{240x^{2}y^{8}z^{2}+52224x^{2}y^{4}z^{6}+2424832x^{2}z^{10}-24xy^{10}z-3328xy^{6}z^{5}-720896xy^{2}z^{9}+y^{12}-512y^{8}z^{4}+28672y^{4}z^{8}+16777216z^{12}}{z^{4}(96x^{2}y^{4}z^{2}-1024x^{2}z^{6}-16xy^{6}z+1280xy^{2}z^{5}+y^{8}-256y^{4}z^{4})}$ |
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.dl.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.dm.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.dp.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.dq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.eb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.ec.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.ef.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.eg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.l.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.w.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.bl.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.br.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.72.5.dt.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.du.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.dx.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.dy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.if.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.ig.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.ij.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.ik.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.72.5.iv.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.72.5.iw.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.iz.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.ja.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.108.7.gf.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.gm.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.1.uk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.um.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.uo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.uq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.vq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.vs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.vu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.vw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.bya.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bye.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bys.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.byw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bzw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bzy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.caa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cac.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cam.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cao.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.caq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cas.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.180.13.bwi.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.cha.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.im.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.io.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.iq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.is.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.js.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.ju.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.jw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.jy.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.biu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.biw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.biy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bja.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjo.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bjq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bkq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bks.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bku.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bkw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.blg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bli.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.blk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.blm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.23.bo.1 | $168$ | $8$ | $8$ | $23$ | $?$ | not computed |
| 264.72.1.ii.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.ik.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.im.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.io.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.jo.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.jq.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.js.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.ju.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.5.biu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.biw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.biy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bja.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjo.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bjq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bkq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bks.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bku.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bkw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.blg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bli.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.blk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.blm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.1.im.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.io.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.iq.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.is.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.js.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.ju.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.jw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.jy.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.5.biu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.biw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.biy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bja.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjo.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bjq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bkq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bks.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bku.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bkw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.blg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bli.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.blk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.blm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |