Invariants
Level: | $60$ | $\SL_2$-level: | $10$ | Newform level: | $900$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
Elliptic points: | $0$ 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: | 10K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.338 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&25\\50&29\end{bmatrix}$, $\begin{bmatrix}23&50\\46&57\end{bmatrix}$, $\begin{bmatrix}37&35\\22&13\end{bmatrix}$, $\begin{bmatrix}49&0\\14&49\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 60-isogeny field degree: | $8$ |
Cyclic 60-torsion field degree: | $128$ |
Full 60-torsion field degree: | $30720$ |
Jacobian
Conductor: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 900.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 y z - 3 z^{2} + w^{2} $ |
$=$ | $5 x^{2} + y^{2} + 3 y z + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 45 x^{4} + 5 x^{2} y^{2} - 6 x^{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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 3x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{19683y^{18}+4645188y^{16}w^{2}+373320900y^{14}w^{4}+10823941224y^{12}w^{6}+49777114356y^{10}w^{8}+76709059992y^{8}w^{10}+43419224232y^{6}w^{12}+22815533520y^{4}w^{14}-2025399852y^{2}w^{16}+600677490214692z^{18}-1201354984995840z^{16}w^{2}+1081219133540700z^{14}w^{4}-583778674368000z^{12}w^{6}+212852144250000z^{10}w^{8}-55324684800000z^{8}w^{10}+10223221950000z^{6}w^{12}-1235736000000z^{4}w^{14}+76381875000z^{2}w^{16}-694570568w^{18}}{w^{2}(2187y^{16}-26244y^{14}w^{2}+146286y^{12}w^{4}-505440y^{10}w^{6}+1223613y^{8}w^{8}-2239236y^{6}w^{10}+3296622y^{4}w^{12}-4152848y^{2}w^{14}-2187z^{16}+37908z^{14}w^{2}-297918z^{12}w^{4}+1401300z^{10}w^{6}+259287750z^{8}w^{8}-341977500z^{6}w^{10}+184256250z^{4}w^{12}-47337500z^{2}w^{14}+4765625w^{16})}$ |
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 |
---|---|---|---|---|---|---|
20.36.0.a.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
30.36.1.f.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
60.24.1.bc.1 | $60$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
60.36.0.e.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 |
---|---|---|---|---|---|---|
60.144.5.fp.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.ft.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.ks.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.ku.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.pd.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.pf.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.pz.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.qd.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.216.13.gm.1 | $60$ | $3$ | $3$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
60.288.13.nq.2 | $60$ | $4$ | $4$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
60.360.13.bv.1 | $60$ | $5$ | $5$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
120.144.5.bok.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.bpm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.dea.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.deo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ehn.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eib.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.enj.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eol.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
300.360.13.bc.1 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |