Invariants
| Level: | $30$ | $\SL_2$-level: | $6$ | Newform level: | $900$ | ||
| Index: | $18$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $1 = 1 + \frac{ 18 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $6^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ 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: | 6C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.18.1.1 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}16&11\\13&16\end{bmatrix}$, $\begin{bmatrix}20&9\\3&28\end{bmatrix}$, $\begin{bmatrix}28&15\\9&16\end{bmatrix}$ |
| 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: | $7680$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 900.2.a.g |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 125 $ |
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 18 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{5^3}\cdot\frac{(y^{2}+375z^{2})^{3}}{z^{2}(y^{2}-125z^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 10.6.0.a.1 | $10$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.9.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 10.6.0.a.1 | $10$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 30.6.1.a.1 | $30$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.36.1.g.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.36.1.i.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.36.1.k.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.36.1.m.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.90.7.f.1 | $30$ | $5$ | $5$ | $7$ | $3$ | $1^{6}$ |
| 30.108.7.c.1 | $30$ | $6$ | $6$ | $7$ | $1$ | $1^{6}$ |
| 30.180.13.cs.1 | $30$ | $10$ | $10$ | $13$ | $4$ | $1^{12}$ |
| 60.36.1.bi.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.bw.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.ck.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.cy.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.2.q.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.s.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.bk.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.bm.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.cm.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.cn.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dc.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dd.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.du.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dv.1 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.36.2.ek.1 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.36.2.el.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.fa.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.fc.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.fi.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.fk.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 90.54.4.a.1 | $90$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 90.162.10.c.1 | $90$ | $9$ | $9$ | $10$ | $?$ | not computed |
| 120.36.1.ea.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.ed.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.fw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.fz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.hs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.hv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.jo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.jr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.2.bx.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.cd.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ev.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.fb.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ia.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.id.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jw.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.jz.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.lu.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.lx.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.nq.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.nt.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.pm.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ps.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.qk.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.qq.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.36.1.i.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.36.1.j.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.36.1.m.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.36.1.n.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.144.10.e.1 | $210$ | $8$ | $8$ | $10$ | $?$ | not computed |
| 330.36.1.e.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.36.1.f.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.36.1.i.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.36.1.j.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.216.16.a.1 | $330$ | $12$ | $12$ | $16$ | $?$ | not computed |