Invariants
| Level: | $230$ | $\SL_2$-level: | $10$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2^{2}\cdot5^{2}\cdot10^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| 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: | 10F0 |
Level structure
| $\GL_2(\Z/230\Z)$-generators: | $\begin{bmatrix}4&55\\133&126\end{bmatrix}$, $\begin{bmatrix}45&4\\218&61\end{bmatrix}$, $\begin{bmatrix}68&223\\47&14\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 10.36.0.a.2 for the level structure with $-I$) |
| Cyclic 230-isogeny field degree: | $24$ |
| Cyclic 230-torsion field degree: | $1056$ |
| Full 230-torsion field degree: | $10686720$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 42 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^5}\cdot\frac{x^{36}(x^{12}+32x^{11}y+416x^{10}y^{2}+2880x^{9}y^{3}+11520x^{8}y^{4}+18432x^{7}y^{5}-65536x^{6}y^{6}-442368x^{5}y^{7}-983040x^{4}y^{8}-655360x^{3}y^{9}+1048576x^{2}y^{10}+2097152xy^{11}+1048576y^{12})^{3}}{y^{5}x^{46}(x+2y)^{5}(x+4y)^{10}(x^{2}+2xy-4y^{2})(x^{2}+12xy+16y^{2})^{2}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $24$ | $12$ | $0$ | $0$ |
| 115.24.0-5.a.1.1 | $115$ | $3$ | $3$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 115.24.0-5.a.1.1 | $115$ | $3$ | $3$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 230.144.1-10.a.1.2 | $230$ | $2$ | $2$ | $1$ |
| 230.144.1-10.b.1.2 | $230$ | $2$ | $2$ | $1$ |
| 230.144.1-230.c.1.2 | $230$ | $2$ | $2$ | $1$ |
| 230.144.1-230.d.1.3 | $230$ | $2$ | $2$ | $1$ |
| 230.360.4-10.a.1.2 | $230$ | $5$ | $5$ | $4$ |