Invariants
| Level: | $30$ | $\SL_2$-level: | $30$ | Newform level: | $20$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot30$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $8$ 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: | 30C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.36.1.2 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}8&27\\15&29\end{bmatrix}$, $\begin{bmatrix}21&5\\29&18\end{bmatrix}$, $\begin{bmatrix}25&16\\23&7\end{bmatrix}$, $\begin{bmatrix}25&28\\23&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 30-isogeny field degree: | $12$ |
| Cyclic 30-torsion field degree: | $96$ |
| Full 30-torsion field degree: | $3840$ |
Jacobian
| Conductor: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} + 4x + 4 $ |
Rational points
This modular curve has rational points, including 2 rational_cusps and 2 known non-cuspidal non-CM points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(0:1:0)$, $(-1:0:1)$ | ||
| 1369.a2 | no | $4096$ | $= 2^{12}$ | $8.318$ | $(0:-2:1)$, $(0:2:1)$ |
| 1369.a1 | no | $38477541376$ | $= 2^{12} \cdot 211^{3}$ | $24.373$ | $(4:-10:1)$, $(4:10:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{12x^{2}y^{10}-305x^{2}y^{8}z^{2}+2540x^{2}y^{6}z^{4}-2151x^{2}y^{4}z^{6}-7772x^{2}y^{2}z^{8}+3481x^{2}z^{10}+54xy^{10}z-868xy^{8}z^{3}+1375xy^{6}z^{5}+13674xy^{4}z^{7}-19435xy^{2}z^{9}-1728xz^{11}+y^{12}+52y^{10}z^{2}+28y^{8}z^{4}-4395y^{6}z^{6}+16603y^{4}z^{8}-9925y^{2}z^{10}-5084z^{12}}{z^{10}(2x^{2}-xz+y^{2}-3z^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 6.6.0.c.1 | $6$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.6.0.c.1 | $6$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 15.18.0.a.1 | $15$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 30.72.1.q.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.72.1.q.2 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.72.1.r.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.72.1.r.2 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.72.5.d.1 | $30$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 30.72.5.i.1 | $30$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 30.72.5.be.1 | $30$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 30.72.5.bf.1 | $30$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 30.72.5.bn.1 | $30$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 30.72.5.bn.2 | $30$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 30.72.5.bo.1 | $30$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 30.72.5.bo.2 | $30$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 30.108.5.g.1 | $30$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 30.180.11.q.1 | $30$ | $5$ | $5$ | $11$ | $2$ | $1^{8}\cdot2$ |
| 60.72.1.fq.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.fq.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.ft.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.ft.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.5.l.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.72.5.ba.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.72.5.dk.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.72.5.dn.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 60.72.5.ej.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.ej.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.em.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.em.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.144.7.ul.1 | $60$ | $4$ | $4$ | $7$ | $1$ | $1^{6}$ |
| 90.108.7.bf.1 | $90$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 90.324.19.bj.1 | $90$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.1.ss.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ss.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.sy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.sy.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.te.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.te.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.tk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.tk.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.kc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.kl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.mk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.mt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bev.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bfb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bfh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bfn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckd.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckj.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckp.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckv.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 150.180.11.c.1 | $150$ | $5$ | $5$ | $11$ | $?$ | not computed |
| 210.72.1.ca.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.72.1.ca.2 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.72.1.cb.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.72.1.cb.2 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.72.5.dr.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.72.5.ds.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.72.5.dz.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.72.5.ea.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.72.5.eh.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.72.5.eh.2 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.72.5.ei.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.72.5.ei.2 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.288.23.t.1 | $210$ | $8$ | $8$ | $23$ | $?$ | not computed |
| 330.72.1.bw.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.72.1.bw.2 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.72.1.bx.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.72.1.bx.2 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.72.5.dr.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.72.5.ds.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.72.5.dz.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.72.5.ea.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.72.5.eh.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.72.5.eh.2 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.72.5.ei.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.72.5.ei.2 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |