Invariants
| Level: | $30$ | $\SL_2$-level: | $10$ | Newform level: | $180$ | ||
| Index: | $30$ | $\PSL_2$-index: | $30$ | ||||
| Genus: | $1 = 1 + \frac{ 30 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $10^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 10E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.30.1.1 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}10&17\\19&0\end{bmatrix}$, $\begin{bmatrix}25&26\\21&25\end{bmatrix}$, $\begin{bmatrix}27&10\\25&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 30-isogeny field degree: | $24$ |
| Cyclic 30-torsion field degree: | $192$ |
| Full 30-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 180.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 12x - 11 $ |
Rational points
This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 30 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^5}\cdot\frac{15x^{2}y^{8}+4860x^{2}y^{6}z^{2}+4107186x^{2}y^{4}z^{4}-5297640084x^{2}y^{2}z^{6}+121801494231x^{2}z^{8}-105xy^{8}z-181764xy^{6}z^{3}-60243102xy^{4}z^{5}+27747125100xy^{2}z^{7}-869140400481xz^{9}+y^{10}-660y^{8}z^{2}+721710y^{6}z^{4}+894211812y^{4}z^{6}-77966350983y^{2}z^{8}-966146983416z^{10}}{z^{3}(9x^{2}y^{4}z+729x^{2}y^{2}z^{3}-6561x^{2}z^{5}-xy^{6}-63xy^{4}z^{2}+6561xz^{6}+5y^{6}z-72y^{4}z^{3}-9477y^{2}z^{5}+72171z^{7})}$ |
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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 30.60.3.a.1 | $30$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 30.60.3.c.1 | $30$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 30.60.3.l.1 | $30$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 30.60.3.m.1 | $30$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 30.90.3.b.1 | $30$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 30.90.4.i.1 | $30$ | $3$ | $3$ | $4$ | $2$ | $1^{3}$ |
| 30.120.8.l.1 | $30$ | $4$ | $4$ | $8$ | $0$ | $1^{7}$ |
| 60.60.3.c.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.60.3.i.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.60.3.bj.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.60.3.bm.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.120.6.d.1 | $60$ | $4$ | $4$ | $6$ | $2$ | $1^{5}$ |
| 120.60.3.c.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.l.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.ba.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.bj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.er.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.ex.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.fd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.fj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 150.150.9.b.1 | $150$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 210.60.3.bg.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.60.3.bh.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.60.3.bj.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.60.3.bk.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.240.18.t.1 | $210$ | $8$ | $8$ | $18$ | $?$ | not computed |
| 330.60.3.bg.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.60.3.bh.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.60.3.bj.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.60.3.bk.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |