Invariants
| Level: | $30$ | $\SL_2$-level: | $10$ | Newform level: | $100$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 30.48.1.32 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}1&15\\4&7\end{bmatrix}$, $\begin{bmatrix}4&25\\1&12\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 10.24.1.b.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: | $2^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 100.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 1033x - 12438 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-18:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5}\cdot\frac{3640x^{2}y^{6}+384408525000x^{2}y^{4}z^{2}+122261047308125000x^{2}y^{2}z^{4}+4471838474273671875000x^{2}z^{6}+4649440xy^{6}z+36733105500000xy^{4}z^{3}+5899143220042500000xy^{2}z^{5}+163002312183380312500000xz^{7}+y^{8}+2361868060y^{6}z^{2}+2272337493493750y^{4}z^{4}+147217083015393437500y^{2}z^{6}+1485165953636176181640625z^{8}}{y^{2}(x^{2}y^{4}+2750x^{2}y^{2}z^{2}-15625x^{2}z^{4}-74xy^{4}z-51625xy^{2}z^{3}+296875xz^{5}+1319y^{4}z^{2}-1923375y^{2}z^{4}+10796875z^{6})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.24.0-5.a.2.1 | $15$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 30.144.1-10.b.2.2 | $30$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 30.240.5-10.e.1.2 | $30$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.192.5-20.e.1.3 | $60$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.144.5-30.w.1.1 | $30$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.192.5-30.e.1.6 | $30$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 150.240.5-50.b.2.1 | $150$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 210.384.13-70.c.1.6 | $210$ | $8$ | $8$ | $13$ | $?$ | not computed |