Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot8$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ 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 $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.12.1.12 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&2\\8&21\end{bmatrix}$, $\begin{bmatrix}7&10\\16&15\end{bmatrix}$, $\begin{bmatrix}13&17\\6&23\end{bmatrix}$, $\begin{bmatrix}19&15\\6&1\end{bmatrix}$, $\begin{bmatrix}23&15\\12&1\end{bmatrix}$ |
| 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: | $6144$ |
Jacobian
| Conductor: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 9x $ |
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 |
|---|
| $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^4}\cdot\frac{2268x^{2}z^{2}+720xy^{2}z+64y^{4}-729z^{4}}{z^{2}x^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.24.1.a.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.h.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.m.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.p.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.cy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.cz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.da.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.db.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dh.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.di.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dj.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.do.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dp.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dr.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dw.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dx.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.dz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.ei.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.el.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.em.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.ep.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.3.h.1 | $24$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.bv.1 | $24$ | $4$ | $4$ | $3$ | $0$ | $1^{2}$ |
| 120.24.1.eg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.eh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.ek.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.el.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.ga.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gt.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.gz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.ha.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.hb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.io.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.ip.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.is.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.24.1.it.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.60.5.dz.1 | $120$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 120.72.5.vv.1 | $120$ | $6$ | $6$ | $5$ | $?$ | not computed |
| 120.120.9.vj.1 | $120$ | $10$ | $10$ | $9$ | $?$ | not computed |
| 168.24.1.du.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.dv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.dy.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.dz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fe.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.ff.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fs.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.ft.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.fv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.ga.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.gb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.gc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.gd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.he.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.hf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.hi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.24.1.hj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.7.dv.1 | $168$ | $8$ | $8$ | $7$ | $?$ | not computed |
| 168.252.19.ln.1 | $168$ | $21$ | $21$ | $19$ | $?$ | not computed |
| 264.24.1.du.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.dv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.dy.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.dz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fe.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.ff.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fk.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fl.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fs.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.ft.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.fv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.ga.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.gb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.gc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.gd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.he.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.hf.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.hi.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.24.1.hj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.144.11.dv.1 | $264$ | $12$ | $12$ | $11$ | $?$ | not computed |
| 312.24.1.du.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.dv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.dy.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.dz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fe.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.ff.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fl.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fm.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fs.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.ft.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fu.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.fv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.ga.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.gb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.gc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.gd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.he.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.hf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.hi.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.24.1.hj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.168.13.hn.1 | $312$ | $14$ | $14$ | $13$ | $?$ | not computed |