Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot8$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.12.0.33 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&3\\18&19\end{bmatrix}$, $\begin{bmatrix}19&16\\0&7\end{bmatrix}$, $\begin{bmatrix}19&21\\20&13\end{bmatrix}$, $\begin{bmatrix}21&13\\4&15\end{bmatrix}$, $\begin{bmatrix}21&13\\8&3\end{bmatrix}$ |
| 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: | $6144$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 18 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3^4}\cdot\frac{(3x+2y)^{12}(3x^{2}-2y^{2})^{3}(3x^{2}+2y^{2})^{3}}{y^{4}x^{8}(3x+2y)^{12}}$ |
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 |
|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.24.0.dg.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dh.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.di.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dj.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.do.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dp.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dq.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dr.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dw.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dx.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dy.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.dz.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.ee.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.ef.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.eg.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0.eh.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.1.a.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.f.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.m.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.o.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.z.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ba.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.bd.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.be.1 | $24$ | $2$ | $2$ | $1$ |
| 24.36.0.ce.1 | $24$ | $3$ | $3$ | $0$ |
| 24.48.3.ck.1 | $24$ | $4$ | $4$ | $3$ |
| 72.324.22.jc.1 | $72$ | $27$ | $27$ | $22$ |
| 120.24.0.ms.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mt.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mu.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mv.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.na.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nb.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nc.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nd.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ni.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nj.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nk.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nl.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nq.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nr.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ns.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nt.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.pg.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ph.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pk.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pl.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pw.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.px.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.qa.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.qb.1 | $120$ | $2$ | $2$ | $1$ |
| 120.60.4.hq.1 | $120$ | $5$ | $5$ | $4$ |
| 120.72.3.gqs.1 | $120$ | $6$ | $6$ | $3$ |
| 120.120.7.boc.1 | $120$ | $10$ | $10$ | $7$ |
| 168.24.0.mm.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mn.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mo.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mp.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mu.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mv.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mw.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.mx.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nc.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nd.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.ne.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nf.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nk.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nl.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nm.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.nn.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.1.ml.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mm.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mp.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mq.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.nb.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.nc.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.nf.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.ng.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.7.ik.1 | $168$ | $8$ | $8$ | $7$ |
| 168.252.14.k.1 | $168$ | $21$ | $21$ | $14$ |
| 168.336.21.bnk.1 | $168$ | $28$ | $28$ | $21$ |
| 264.24.0.mm.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.mn.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.mo.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.mp.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.mu.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.mv.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.mw.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.mx.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.nc.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.nd.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.ne.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.nf.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.nk.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.nl.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.nm.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0.nn.1 | $264$ | $2$ | $2$ | $0$ |
| 264.24.1.mm.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mn.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mq.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mr.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.nc.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.nd.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.ng.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.nh.1 | $264$ | $2$ | $2$ | $1$ |
| 264.144.11.ic.1 | $264$ | $12$ | $12$ | $11$ |
| 312.24.0.ms.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.mt.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.mu.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.mv.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.na.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nb.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nc.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nd.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.ni.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nj.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nk.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nl.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nq.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nr.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.ns.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0.nt.1 | $312$ | $2$ | $2$ | $0$ |
| 312.24.1.mm.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mn.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mq.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mr.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.nc.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.nd.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.ng.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.nh.1 | $312$ | $2$ | $2$ | $1$ |
| 312.168.11.xc.1 | $312$ | $14$ | $14$ | $11$ |