Invariants
Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $1800$ | ||
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: | 60.36.1.106 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}29&14\\47&49\end{bmatrix}$, $\begin{bmatrix}35&12\\24&55\end{bmatrix}$, $\begin{bmatrix}47&50\\26&13\end{bmatrix}$, $\begin{bmatrix}49&18\\3&23\end{bmatrix}$ |
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: | $2^{3}\cdot3^{2}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 1800.2.a.m |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} + 2 x z + z^{2} + w^{2} $ |
$=$ | $4 x^{2} - 13 x z - 2 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 225 x^{4} - 15 x^{2} z^{2} + y^{2} z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{15}{2}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{15}{2}z$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{54xz^{6}w^{2}-54xz^{4}w^{4}+12xz^{2}w^{6}-2xw^{8}+27z^{9}-27z^{7}w^{2}-6z^{3}w^{6}+4zw^{8}}{z^{3}(4xz^{3}w^{2}+6xzw^{4}-z^{6}-2z^{4}w^{2}+w^{6})}$ |
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 |
---|---|---|---|---|---|---|
12.18.0.b.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.18.0.j.1 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.18.1.h.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.bi.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.dx.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.gf.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.gl.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.ls.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.lz.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.ma.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.72.3.mh.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
60.180.13.lr.1 | $60$ | $5$ | $5$ | $13$ | $5$ | $1^{12}$ |
60.216.13.oe.1 | $60$ | $6$ | $6$ | $13$ | $5$ | $1^{12}$ |
60.360.25.cap.1 | $60$ | $10$ | $10$ | $25$ | $9$ | $1^{24}$ |
120.72.3.kv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.yy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bnw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bpm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ddl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dfi.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dfp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dhm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
180.108.5.bi.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
180.324.21.x.1 | $180$ | $9$ | $9$ | $21$ | $?$ | not computed |