Invariants
Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $1200$ | ||
Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12L1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.1.20 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&32\\1&3\end{bmatrix}$, $\begin{bmatrix}13&44\\22&19\end{bmatrix}$, $\begin{bmatrix}41&46\\11&17\end{bmatrix}$, $\begin{bmatrix}45&34\\44&15\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 60-isogeny field degree: | $48$ |
Cyclic 60-torsion field degree: | $768$ |
Full 60-torsion field degree: | $61440$ |
Jacobian
Conductor: | $2^{4}\cdot3\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 1200.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 y^{2} + 2 y z + y w + 2 z^{2} + 2 z w + 2 w^{2} $ |
$=$ | $3 x^{2} - y^{2} - y z + 2 y w - z^{2} + 4 z w - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 3 x^{2} y z - 15 x^{2} z^{2} + 3 y^{2} z^{2} + 900 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle 5z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -3^3\,\frac{1536yz^{8}-768yz^{7}w-3264yz^{6}w^{2}-5376yz^{5}w^{3}+96yz^{4}w^{4}+1488yz^{3}w^{5}+828yz^{2}w^{6}+288yzw^{7}+33yw^{8}+768z^{9}-3456z^{8}w-4416z^{7}w^{2}-4512z^{6}w^{3}+3888z^{5}w^{4}+5160z^{4}w^{5}+2460z^{3}w^{6}+342z^{2}w^{7}+30zw^{8}+10w^{9}}{w^{6}(24yz^{2}-12yzw-3yw^{2}+12z^{3}-18z^{2}w-18zw^{2}+2w^{3})}$ |
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 |
---|---|---|---|---|---|---|
6.18.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.18.0.j.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.18.1.i.1 | $60$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
60.72.3.i.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.dj.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.fo.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.fu.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.ia.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.ig.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.iq.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.iw.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.180.13.la.1 | $60$ | $5$ | $5$ | $13$ | $4$ | $1^{12}$ |
60.216.13.nn.1 | $60$ | $6$ | $6$ | $13$ | $5$ | $1^{12}$ |
60.360.25.bzy.1 | $60$ | $10$ | $10$ | $25$ | $7$ | $1^{24}$ |
120.72.3.ke.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.vi.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bjh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bkx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.chh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.cix.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.clp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.cnf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
180.108.5.p.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
180.108.5.z.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
180.324.21.g.1 | $180$ | $9$ | $9$ | $21$ | $?$ | not computed |