Invariants
| Level: | $60$ | $\SL_2$-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: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot30^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.412 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}14&15\\33&23\end{bmatrix}$, $\begin{bmatrix}17&15\\27&4\end{bmatrix}$, $\begin{bmatrix}39&50\\26&3\end{bmatrix}$, $\begin{bmatrix}53&5\\4&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + y w $ |
| $=$ | $5 y^{2} + 2 y w - 3 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} - 6 x^{2} z^{2} - 3 y^{2} z^{2} + 9 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{1708593750yz^{16}w+6264843750yz^{14}w^{3}+21740906250yz^{12}w^{5}+50177981250yz^{10}w^{7}+31569243750yz^{8}w^{9}+32664890250yz^{6}w^{11}+2447671500yz^{4}w^{13}-253083840yz^{2}w^{15}-4376384yw^{17}-284765625z^{18}-1879453125z^{16}w^{2}-11390625000z^{14}w^{4}-14144878125z^{12}w^{6}-24959137500z^{10}w^{8}-25000214625z^{8}w^{10}+11568316725z^{6}w^{12}-1568527200z^{4}w^{14}-133433904z^{2}w^{16}+515968w^{18}}{w^{3}(170859375yz^{14}+877078125yz^{12}w^{2}-621928125yz^{10}w^{4}+33665625yz^{8}w^{6}+65188125yz^{6}w^{8}-19269225yz^{4}w^{10}+2021745yz^{2}w^{12}-68381yw^{14}-478406250z^{14}w+95681250z^{12}w^{3}+312558750z^{10}w^{5}-215763750z^{8}w^{7}+57233250z^{6}w^{9}-6634170z^{4}w^{11}+222954z^{2}w^{13}+8062w^{15})}$ |
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 |
|---|---|---|---|---|---|---|
| 30.36.1.q.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.0.j.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.36.0.cg.1 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.144.9.bc.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.bf.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.cn.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.cr.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.ix.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.iz.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.je.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 60.144.9.jg.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 60.216.9.bi.1 | $60$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 60.288.13.sl.1 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.360.21.db.1 | $60$ | $5$ | $5$ | $21$ | $4$ | $1^{8}\cdot2^{6}$ |
| 120.144.9.ixg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.iyb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.jth.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.juj.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tcr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tdf.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tev.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.tfj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 180.216.13.hn.2 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 300.360.21.t.1 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |