Invariants
| Level: | $30$ | $\SL_2$-level: | $30$ | Newform level: | $20$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{2}\cdot30^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $16$ 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: | 30D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.72.1.23 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}5&12\\18&25\end{bmatrix}$, $\begin{bmatrix}7&2\\5&23\end{bmatrix}$, $\begin{bmatrix}25&21\\27&4\end{bmatrix}$ |
| 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: | $1920$ |
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} - x $ |
Rational points
This modular curve has rational points, including 2 rational_cusps and 1 known non-cuspidal non-CM point. 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 | $\infty$ | $0.000$ | $(0:0:1)$, $(0:1:0)$ | ||
| 1369.a1 | no | $38477541376$ | $= 2^{12} \cdot 211^{3}$ | $24.373$ | $(-1:-1:1)$, $(-1:1:1)$, $(1:-1:1)$, $(1:1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{6x^{2}y^{22}+2951x^{2}y^{20}z^{2}+382620x^{2}y^{18}z^{4}-4383x^{2}y^{16}z^{6}-1231220040x^{2}y^{14}z^{8}-26220899967x^{2}y^{12}z^{10}+190429667778x^{2}y^{10}z^{12}+4846191398815x^{2}y^{8}z^{14}-39753417969682x^{2}y^{6}z^{16}+107482910156160x^{2}y^{4}z^{18}-121154785156266x^{2}y^{2}z^{20}+48736572265624x^{2}z^{22}+15xy^{22}z+4437xy^{20}z^{3}+193870xy^{18}z^{5}+564030xy^{16}z^{7}+1845143550xy^{14}z^{9}+71396302170xy^{12}z^{11}+201416002554xy^{10}z^{13}-9027099615930xy^{8}z^{15}+46472167967895xy^{6}z^{17}-95947265625075xy^{4}z^{19}+88348388671860xy^{2}z^{21}-30120849609376xz^{23}+y^{24}+743y^{22}z^{2}+189487y^{20}z^{4}+19358510y^{18}z^{6}+615617670y^{16}z^{8}+5479448358y^{14}z^{10}-130615028786y^{12}z^{12}-1463623021340y^{10}z^{14}+17724609383250y^{8}z^{16}-56030273436505y^{6}z^{18}+69732666015729y^{4}z^{20}-30120849609358y^{2}z^{22}+z^{24}}{z^{2}y^{6}(4x^{2}y^{14}-15x^{2}y^{12}z^{2}-6x^{2}y^{10}z^{4}+49x^{2}y^{8}z^{6}-27x^{2}y^{4}z^{10}-10x^{2}y^{2}z^{12}-x^{2}z^{14}+6xy^{14}z-4xy^{12}z^{3}-36xy^{10}z^{5}+42xy^{8}z^{7}+20xy^{6}z^{9}-18xy^{4}z^{11}-9xy^{2}z^{13}-xz^{15}+y^{16}-10y^{14}z^{2}+41y^{12}z^{4}-14y^{10}z^{6}-70y^{8}z^{8}+12y^{6}z^{10}+35y^{4}z^{12}+11y^{2}z^{14}+z^{16})}$ |
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 |
|---|---|---|---|---|---|---|
| 15.36.0.a.2 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.36.0.f.1 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.36.1.q.1 | $30$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 30.144.9.e.1 | $30$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 30.144.9.m.1 | $30$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 30.144.9.bn.2 | $30$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 30.144.9.bq.1 | $30$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 30.216.9.g.1 | $30$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 30.360.21.l.1 | $30$ | $5$ | $5$ | $21$ | $2$ | $1^{8}\cdot2^{6}$ |
| 60.144.9.ba.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.da.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.iu.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.jp.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.288.13.sr.1 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 90.216.13.bz.2 | $90$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.9.iuo.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.iwl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.kkf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.klv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tag.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tbp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tgz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tib.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 150.360.21.f.1 | $150$ | $5$ | $5$ | $21$ | $?$ | not computed |
| 210.144.9.kc.1 | $210$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 210.144.9.kd.2 | $210$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 210.144.9.ks.2 | $210$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 210.144.9.kt.1 | $210$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 330.144.9.kc.1 | $330$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 330.144.9.kd.1 | $330$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 330.144.9.ks.1 | $330$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 330.144.9.kt.1 | $330$ | $2$ | $2$ | $9$ | $?$ | not computed |