Invariants
| Level: | $30$ | $\SL_2$-level: | $10$ | Newform level: | $900$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.48.1.7 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}11&10\\17&11\end{bmatrix}$, $\begin{bmatrix}16&25\\27&22\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.24.1.i.2 for the level structure with $-I$) |
| Cyclic 30-isogeny field degree: | $12$ |
| Cyclic 30-torsion field degree: | $96$ |
| Full 30-torsion field degree: | $2880$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 900.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 9300x + 345125 $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3\cdot5}\cdot\frac{10920x^{2}y^{6}-31137090525000x^{2}y^{4}z^{2}+267384910462869375000x^{2}y^{2}z^{4}-264057590067386050546875000x^{2}z^{6}-41866800xy^{6}z+8988418817550000xy^{4}z^{3}-39239048487624581250000xy^{2}z^{5}+29403485776484044319531250000xz^{7}-y^{8}+63812293500y^{6}z^{2}-1665491314483968750y^{4}z^{4}+2936645508569150742187500y^{2}z^{6}-818417507752779764416259765625z^{8}}{y^{2}(x^{2}y^{4}-74250x^{2}y^{2}z^{2}-11390625x^{2}z^{4}+220xy^{4}z-4033125xy^{2}z^{3}-626484375xz^{5}+11650y^{4}z^{2}+471487500y^{2}z^{4}+71476171875z^{6})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 5.24.0-5.a.2.2 | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.24.0-5.a.2.2 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.144.1-30.i.2.4 | $30$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
| 30.144.5-30.ba.1.2 | $30$ | $3$ | $3$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 30.192.5-30.g.2.2 | $30$ | $4$ | $4$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 30.240.5-30.m.1.4 | $30$ | $5$ | $5$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.192.5-60.cg.1.7 | $60$ | $4$ | $4$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 150.240.5-150.e.2.3 | $150$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 210.384.13-210.r.1.5 | $210$ | $8$ | $8$ | $13$ | $?$ | not computed |