Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.0.440 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&16\\8&13\end{bmatrix}$, $\begin{bmatrix}11&12\\0&11\end{bmatrix}$, $\begin{bmatrix}19&2\\8&9\end{bmatrix}$, $\begin{bmatrix}21&22\\8&1\end{bmatrix}$, $\begin{bmatrix}23&12\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.i.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $1536$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 122 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}+240x^{6}y^{2}+2144x^{4}y^{4}+3840x^{2}y^{6}+256y^{8})^{3}}{y^{2}x^{26}(x-2y)^{8}(x+2y)^{8}(x^{2}+4y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-4.b.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-8.n.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.96.0-8.j.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-8.j.2.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-8.k.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-8.k.2.7 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-8.l.1.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-8.l.2.2 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.ba.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.ba.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bb.1.14 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bb.2.10 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bc.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bc.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-8.h.1.12 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-8.p.1.6 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bu.1.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bv.1.10 | $24$ | $2$ | $2$ | $1$ |
| 24.144.4-24.ch.1.25 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.cl.1.43 | $24$ | $4$ | $4$ | $3$ |
| 48.96.0-16.d.1.9 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.d.2.12 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.d.1.23 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.d.2.5 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-16.a.1.9 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.a.2.12 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.a.1.24 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.a.2.9 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.b.1.9 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.b.2.12 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.b.1.24 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.b.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.96.2-16.d.1.11 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2-16.d.2.10 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2-48.d.1.23 | $48$ | $2$ | $2$ | $2$ |
| 48.96.2-48.d.2.6 | $48$ | $2$ | $2$ | $2$ |
| 120.96.0-40.bb.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.bb.2.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.bc.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.bc.2.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.bd.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.bd.2.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cy.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cy.2.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cz.1.23 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cz.2.21 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.da.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.da.2.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-40.bu.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.bv.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fq.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fr.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.240.8-40.v.1.11 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-40.br.1.24 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-40.ch.1.48 | $120$ | $10$ | $10$ | $15$ |
| 168.96.0-56.z.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.z.2.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.ba.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.ba.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bb.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bb.2.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cw.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cw.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cx.1.18 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cx.2.20 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cy.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cy.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-56.bu.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-56.bv.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fq.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fr.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.384.11-56.bn.1.20 | $168$ | $8$ | $8$ | $11$ |
| 240.96.0-80.f.1.23 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.f.2.8 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.f.1.37 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.f.2.10 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-80.a.1.21 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.a.2.12 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.a.1.40 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.a.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.b.1.23 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.b.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.b.1.40 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.b.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.96.2-80.f.1.22 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-80.f.2.12 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.f.1.38 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.f.2.7 | $240$ | $2$ | $2$ | $2$ |
| 264.96.0-88.z.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.z.2.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.ba.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.ba.2.16 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bb.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bb.2.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cw.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cw.2.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cx.1.22 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cx.2.19 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cy.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.cy.2.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-88.bu.1.15 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-88.bv.1.15 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fq.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.fr.1.9 | $264$ | $2$ | $2$ | $1$ |
| 312.96.0-104.bb.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bb.2.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bc.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bc.2.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bd.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bd.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cy.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cy.2.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cz.1.18 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.cz.2.20 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.da.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.da.2.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-104.bu.1.16 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-104.bv.1.16 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fq.1.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.fr.1.6 | $312$ | $2$ | $2$ | $1$ |