Invariants
| Level: | $6$ | $\SL_2$-level: | $6$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $2^{3}\cdot6^{3}$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 6.48.0.4 |
Level structure
| $\GL_2(\Z/6\Z)$-generators: | $\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/6\Z)$-subgroup: | $S_3$ |
| Contains $-I$: | no $\quad$ (see 6.24.0.a.1 for the level structure with $-I$) |
| Cyclic 6-isogeny field degree: | $1$ |
| Cyclic 6-torsion field degree: | $2$ |
| Full 6-torsion field degree: | $6$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 110 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{24}(x^{2}+12y^{2})^{3}(x^{6}-60x^{4}y^{2}+1200x^{2}y^{4}+192y^{6})^{3}}{y^{6}x^{26}(x-6y)^{2}(x-2y)^{6}(x+2y)^{6}(x+6y)^{2}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X(2)$ | $2$ | $8$ | $4$ | $0$ | $0$ |
| 3.8.0-3.a.1.1 | $3$ | $6$ | $6$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.16.0-6.a.1.1 | $6$ | $3$ | $3$ | $0$ | $0$ |
| 6.24.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 6.24.0-6.a.1.3 | $6$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\mathrm{arith}}(6)$ | $6$ | $3$ | $3$ | $1$ |
| 12.96.0-12.a.1.15 | $12$ | $2$ | $2$ | $0$ |
| 12.96.0-12.a.2.15 | $12$ | $2$ | $2$ | $0$ |
| 12.96.1-12.a.1.12 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.b.1.12 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.c.1.10 | $12$ | $2$ | $2$ | $1$ |
| 12.96.1-12.d.1.10 | $12$ | $2$ | $2$ | $1$ |
| 12.96.2-12.a.1.8 | $12$ | $2$ | $2$ | $2$ |
| 12.96.2-12.a.2.8 | $12$ | $2$ | $2$ | $2$ |
| 18.144.1-18.a.1.1 | $18$ | $3$ | $3$ | $1$ |
| 18.144.4-18.a.1.1 | $18$ | $3$ | $3$ | $4$ |
| 18.144.4-18.b.1.1 | $18$ | $3$ | $3$ | $4$ |
| 24.96.0-24.o.1.31 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.o.2.31 | $24$ | $2$ | $2$ | $0$ |
| 24.96.1-24.bw.1.10 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bx.1.10 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.by.1.20 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1-24.bz.1.20 | $24$ | $2$ | $2$ | $1$ |
| 24.96.2-24.b.1.16 | $24$ | $2$ | $2$ | $2$ |
| 24.96.2-24.b.2.12 | $24$ | $2$ | $2$ | $2$ |
| 30.240.8-30.a.1.4 | $30$ | $5$ | $5$ | $8$ |
| 30.288.7-30.a.1.4 | $30$ | $6$ | $6$ | $7$ |
| 30.480.15-30.a.1.6 | $30$ | $10$ | $10$ | $15$ |
| 42.384.11-42.a.1.7 | $42$ | $8$ | $8$ | $11$ |
| 42.1008.34-42.a.1.8 | $42$ | $21$ | $21$ | $34$ |
| 42.1344.45-42.a.1.8 | $42$ | $28$ | $28$ | $45$ |
| 60.96.0-60.a.1.16 | $60$ | $2$ | $2$ | $0$ |
| 60.96.0-60.a.2.32 | $60$ | $2$ | $2$ | $0$ |
| 60.96.1-60.a.1.17 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.b.1.18 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.c.1.18 | $60$ | $2$ | $2$ | $1$ |
| 60.96.1-60.d.1.13 | $60$ | $2$ | $2$ | $1$ |
| 60.96.2-60.a.1.12 | $60$ | $2$ | $2$ | $2$ |
| 60.96.2-60.a.2.16 | $60$ | $2$ | $2$ | $2$ |
| 66.576.19-66.a.1.8 | $66$ | $12$ | $12$ | $19$ |
| 66.2640.96-66.a.1.3 | $66$ | $55$ | $55$ | $96$ |
| 66.2640.96-66.b.1.6 | $66$ | $55$ | $55$ | $96$ |
| 66.3168.115-66.a.1.7 | $66$ | $66$ | $66$ | $115$ |
| 84.96.0-84.a.1.24 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.a.2.8 | $84$ | $2$ | $2$ | $0$ |
| 84.96.1-84.a.1.16 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.c.1.16 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.d.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.2-84.a.1.14 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-84.a.2.8 | $84$ | $2$ | $2$ | $2$ |
| 120.96.0-120.o.1.32 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.o.2.64 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.dg.1.34 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dh.1.36 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.di.1.38 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.dj.1.27 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-120.b.1.24 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.b.2.32 | $120$ | $2$ | $2$ | $2$ |
| 132.96.0-132.a.1.24 | $132$ | $2$ | $2$ | $0$ |
| 132.96.0-132.a.2.32 | $132$ | $2$ | $2$ | $0$ |
| 132.96.1-132.a.1.17 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.b.1.18 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.c.1.18 | $132$ | $2$ | $2$ | $1$ |
| 132.96.1-132.d.1.17 | $132$ | $2$ | $2$ | $1$ |
| 132.96.2-132.a.1.14 | $132$ | $2$ | $2$ | $2$ |
| 132.96.2-132.a.2.16 | $132$ | $2$ | $2$ | $2$ |
| 156.96.0-156.a.1.8 | $156$ | $2$ | $2$ | $0$ |
| 156.96.0-156.a.2.16 | $156$ | $2$ | $2$ | $0$ |
| 156.96.1-156.a.1.11 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.b.1.8 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.c.1.9 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1-156.d.1.7 | $156$ | $2$ | $2$ | $1$ |
| 156.96.2-156.a.1.8 | $156$ | $2$ | $2$ | $2$ |
| 156.96.2-156.a.2.12 | $156$ | $2$ | $2$ | $2$ |
| 168.96.0-168.o.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.o.2.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.dg.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dh.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.di.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dj.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-168.b.1.12 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.b.2.12 | $168$ | $2$ | $2$ | $2$ |
| 204.96.0-204.a.1.16 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-204.a.2.32 | $204$ | $2$ | $2$ | $0$ |
| 204.96.1-204.a.1.13 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.b.1.18 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.c.1.18 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.d.1.13 | $204$ | $2$ | $2$ | $1$ |
| 204.96.2-204.a.1.12 | $204$ | $2$ | $2$ | $2$ |
| 204.96.2-204.a.2.16 | $204$ | $2$ | $2$ | $2$ |
| 228.96.0-228.a.1.8 | $228$ | $2$ | $2$ | $0$ |
| 228.96.0-228.a.2.16 | $228$ | $2$ | $2$ | $0$ |
| 228.96.1-228.a.1.11 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.b.1.8 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.c.1.9 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.d.1.7 | $228$ | $2$ | $2$ | $1$ |
| 228.96.2-228.a.1.8 | $228$ | $2$ | $2$ | $2$ |
| 228.96.2-228.a.2.12 | $228$ | $2$ | $2$ | $2$ |
| 264.96.0-264.o.1.23 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.o.2.63 | $264$ | $2$ | $2$ | $0$ |
| 264.96.1-264.dg.1.18 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.dh.1.18 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.di.1.35 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.dj.1.35 | $264$ | $2$ | $2$ | $1$ |
| 264.96.2-264.b.1.23 | $264$ | $2$ | $2$ | $2$ |
| 264.96.2-264.b.2.27 | $264$ | $2$ | $2$ | $2$ |
| 276.96.0-276.a.1.24 | $276$ | $2$ | $2$ | $0$ |
| 276.96.0-276.a.2.32 | $276$ | $2$ | $2$ | $0$ |
| 276.96.1-276.a.1.17 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.b.1.19 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.c.1.18 | $276$ | $2$ | $2$ | $1$ |
| 276.96.1-276.d.1.17 | $276$ | $2$ | $2$ | $1$ |
| 276.96.2-276.a.1.14 | $276$ | $2$ | $2$ | $2$ |
| 276.96.2-276.a.2.16 | $276$ | $2$ | $2$ | $2$ |
| 312.96.0-312.o.1.32 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.o.2.16 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-312.dg.1.18 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dh.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.di.1.30 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.dj.1.8 | $312$ | $2$ | $2$ | $1$ |
| 312.96.2-312.b.1.24 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.b.2.16 | $312$ | $2$ | $2$ | $2$ |