Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ 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: | 8N0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.0.42 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4}\cdot\frac{(2x-y)^{48}(1296x^{8}-432x^{6}y^{2}+72x^{4}y^{4}+12x^{2}y^{6}+y^{8})^{3}(1296x^{8}+432x^{6}y^{2}+72x^{4}y^{4}-12x^{2}y^{6}+y^{8})^{3}}{y^{8}x^{8}(2x-y)^{48}(6x^{2}-y^{2})^{4}(6x^{2}+y^{2})^{4}(36x^{4}+y^{4})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0.h.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0.i.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.96.1.a.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.b.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.e.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.f.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.l.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.m.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.n.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.o.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.p.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.q.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.w.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.x.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.3.m.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.n.2 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.p.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.s.2 | $24$ | $2$ | $2$ | $3$ |
| 24.144.8.g.1 | $24$ | $3$ | $3$ | $8$ |
| 24.192.7.f.1 | $24$ | $4$ | $4$ | $7$ |
| 120.96.1.o.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.p.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.y.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.z.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bn.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bo.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bp.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bq.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.by.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ck.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.cl.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.3.ca.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.cb.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.cc.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.cd.1 | $120$ | $2$ | $2$ | $3$ |
| 120.240.16.d.1 | $120$ | $5$ | $5$ | $16$ |
| 120.288.15.f.1 | $120$ | $6$ | $6$ | $15$ |
| 168.96.1.o.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.p.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.y.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.z.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.bn.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.bo.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.bp.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.bq.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.bx.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.by.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ck.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.cl.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.3.bs.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.bt.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.bu.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.bv.2 | $168$ | $2$ | $2$ | $3$ |
| 168.384.23.g.1 | $168$ | $8$ | $8$ | $23$ |
| 264.96.1.o.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.p.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.y.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.z.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.bn.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.bo.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.bp.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.bq.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.bx.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.by.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.ck.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.cl.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.3.bs.1 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.bt.2 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.bu.1 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.bv.2 | $264$ | $2$ | $2$ | $3$ |
| 312.96.1.o.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.p.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.y.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.z.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.bn.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.bo.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.bp.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.bq.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.bx.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.by.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ck.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.cl.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.3.ca.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.cb.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.cc.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.cd.2 | $312$ | $2$ | $2$ | $3$ |