Invariants
| Level: | $30$ | $\SL_2$-level: | $10$ | Newform level: | $900$ | ||
| Index: | $20$ | $\PSL_2$-index: | $20$ | ||||
| Genus: | $1 = 1 + \frac{ 20 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $10^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-27$) |
Other labels
| Cummins and Pauli (CP) label: | 10C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.20.1.1 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}1&19\\9&4\end{bmatrix}$, $\begin{bmatrix}4&1\\1&17\end{bmatrix}$, $\begin{bmatrix}6&25\\25&6\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 30-isogeny field degree: | $72$ |
| Cyclic 30-torsion field degree: | $576$ |
| Full 30-torsion field degree: | $6912$ |
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} + 825x - 9250 $ |
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 20 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3\cdot5^3\,\frac{z(81311040x^{2}y^{11}-4311995832000x^{2}y^{10}z-17225433041478750x^{2}y^{9}z^{2}-28479347232261890625x^{2}y^{8}z^{3}-21901553896629324375000x^{2}y^{7}z^{4}-13414907499313352449218750x^{2}y^{6}z^{5}-3149340135885131712539062500x^{2}y^{5}z^{6}-99803671059726500315185546875x^{2}y^{4}z^{7}-650265282618264402262822265625000x^{2}y^{3}z^{8}+110597944962147652190569427490234375x^{2}y^{2}z^{9}+13671654958585758017498277282714843750x^{2}yz^{10}-938805262485830298322070560455322265625x^{2}z^{11}-753920xy^{12}+189835758000xy^{11}z+356217897076875xy^{10}z^{2}+645616208134800000xy^{9}z^{3}+869587605477122343750xy^{8}z^{4}+810622714115987310937500xy^{7}z^{5}+449537768146793348613281250xy^{6}z^{6}+340462285170371997929296875000xy^{5}z^{7}-41317187989172536565255126953125xy^{4}z^{8}-3015410694734470115077844238281250xy^{3}z^{9}+4855698593971893407503730621337890625xy^{2}z^{10}-639600649659062486750757705688476562500xyz^{11}-2605348163425560433486806678771972656250xz^{12}+3072y^{13}-5008644400y^{12}z+44919381357000y^{11}z^{2}+173427119099634375y^{10}z^{3}+178034559013016718750y^{9}z^{4}+110913816400086111328125y^{8}z^{5}+433139059063631531250000y^{7}z^{6}+3346037189313841431738281250y^{6}z^{7}-12065354265678494984441894531250y^{5}z^{8}-5791320678240194982074560546875000y^{4}z^{9}+413444815911231685707265136718750000y^{3}z^{10}-40764158008428902224787711334228515625y^{2}z^{11}+5026908984746377980563996086120605468750yz^{12}+119922982562256358697287703990936279296875z^{13})}{x^{2}y^{12}-7552838250x^{2}y^{10}z^{2}+932588110949765625x^{2}y^{8}z^{4}-4837647513189367617187500x^{2}y^{6}z^{6}-67938331187216945271972656250x^{2}y^{4}z^{8}-3920223801426136260760986328125000x^{2}y^{2}z^{10}-109238254790651513665651222229003906250x^{2}z^{12}-3950xy^{12}z+5357546921250xy^{10}z^{3}-232955003521293750000xy^{8}z^{5}+340160610700169067832031250xy^{6}z^{7}+27419387760845717750112304687500xy^{4}z^{9}+786048279992514762937598419189453125xy^{2}z^{11}+6018753982803745466320273704528808593750xz^{13}+7065925y^{12}z^{2}-2652664528125000y^{10}z^{4}+41249339980155808593750y^{8}z^{6}-13603300636156494134765625000y^{6}z^{8}-729323050484112039496142578125000y^{4}z^{10}-10876450551173750546620133972167968750y^{2}z^{12}-49263714345724982271169205760955810546875z^{14}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 6.2.0.a.1 | $6$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 6.2.0.a.1 | $6$ | $10$ | $10$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 30.40.1.c.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 30.40.1.d.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 30.40.1.e.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 30.40.1.f.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 30.60.3.d.1 | $30$ | $3$ | $3$ | $3$ | $2$ | $1^{2}$ |
| 30.60.3.e.1 | $30$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 30.60.4.a.1 | $30$ | $3$ | $3$ | $4$ | $3$ | $1^{3}$ |
| 30.60.5.h.1 | $30$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 30.80.5.b.1 | $30$ | $4$ | $4$ | $5$ | $3$ | $1^{4}$ |
| 60.40.1.g.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.40.1.j.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.40.1.m.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.40.1.p.1 | $60$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 60.80.5.c.1 | $60$ | $4$ | $4$ | $5$ | $5$ | $1^{4}$ |
| 120.40.1.y.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.bb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.bk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.bn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.bw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.bz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.ci.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.cl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 150.100.5.a.1 | $150$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 210.40.1.c.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.40.1.d.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.40.1.e.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.40.1.f.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.160.11.a.1 | $210$ | $8$ | $8$ | $11$ | $?$ | not computed |
| 330.40.1.c.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.40.1.d.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.40.1.e.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.40.1.f.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.240.19.a.1 | $330$ | $12$ | $12$ | $19$ | $?$ | not computed |