Invariants
| Level: | $30$ | $\SL_2$-level: | $10$ | Newform level: | $900$ | ||
| 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: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-4$) |
Other labels
| Cummins and Pauli (CP) label: | 10E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.30.1.2 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}5&3\\17&10\end{bmatrix}$, $\begin{bmatrix}23&10\\5&23\end{bmatrix}$, $\begin{bmatrix}23&15\\25&4\end{bmatrix}$ |
| 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: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 900.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 300x - 1375 $ |
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 30 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^5\cdot5^5}\cdot\frac{75x^{2}y^{8}+3037500x^{2}y^{6}z^{2}+320873906250x^{2}y^{4}z^{4}-51734766445312500x^{2}y^{2}z^{6}+148683464637451171875x^{2}z^{8}-2625xy^{8}z-568012500xy^{6}z^{3}-23532461718750xy^{4}z^{5}+1354840092773437500xy^{2}z^{7}-5304812014654541015625xz^{9}+y^{10}-82500y^{8}z^{2}+11276718750y^{6}z^{4}+1746507445312500y^{4}z^{6}-19034753657958984375y^{2}z^{8}-29484466046630859375000z^{10}}{z^{3}(225x^{2}y^{4}z+2278125x^{2}y^{2}z^{3}-2562890625x^{2}z^{5}-xy^{6}-7875xy^{4}z^{2}+12814453125xz^{6}+25y^{6}z-45000y^{4}z^{3}-740390625y^{2}z^{5}+704794921875z^{7})}$ |
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.b.1 | $30$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 30.60.3.d.1 | $30$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 30.60.3.h.1 | $30$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 30.60.3.i.1 | $30$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 30.90.3.a.1 | $30$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 30.90.4.c.1 | $30$ | $3$ | $3$ | $4$ | $2$ | $1^{3}$ |
| 30.120.8.j.1 | $30$ | $4$ | $4$ | $8$ | $3$ | $1^{7}$ |
| 60.60.3.f.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.60.3.l.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.60.3.x.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.60.3.ba.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.120.6.b.1 | $60$ | $4$ | $4$ | $6$ | $5$ | $1^{5}$ |
| 120.60.3.o.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.x.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.bm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.bv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.da.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.dj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.dm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.dv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 150.150.9.a.1 | $150$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 210.60.3.ba.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.60.3.bb.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.60.3.bd.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.60.3.be.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.240.18.s.1 | $210$ | $8$ | $8$ | $18$ | $?$ | not computed |
| 330.60.3.ba.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.60.3.bb.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.60.3.bd.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.60.3.be.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |