Invariants
Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $600$ | ||
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: | $0$ | ||||||
$\Q$-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: | 60.36.1.88 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&8\\53&57\end{bmatrix}$, $\begin{bmatrix}5&38\\41&31\end{bmatrix}$, $\begin{bmatrix}23&50\\23&31\end{bmatrix}$, $\begin{bmatrix}49&8\\1&1\end{bmatrix}$ |
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: | $2^{3}\cdot3\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 600.2.a.h |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + 4 y^{2} + y z + z^{2} + w^{2} $ |
$=$ | $3 x^{2} - 4 y^{2} - y z + y w - z^{2} + 2 z w - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 15 x^{2} y z + 3 x^{2} z^{2} + 15 y^{2} z^{2} + 36 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^{12}}\cdot\frac{374880000yz^{8}+556800000yz^{7}w-289152000yz^{6}w^{2}+220704000yz^{5}w^{3}+215493600yz^{4}w^{4}-87494400yz^{3}w^{5}+60200160yz^{2}w^{6}+8898960yzw^{7}-1666365yw^{8}-107040000z^{9}+257040000z^{8}w-171384000z^{7}w^{2}-169116000z^{6}w^{3}+208825200z^{5}w^{4}-112621800z^{4}w^{5}+8396220z^{3}w^{6}+20222190z^{2}w^{7}-9674790zw^{8}+4691482w^{9}}{w^{6}(660yz^{2}+60yzw-15yw^{2}+420z^{3}+90z^{2}w-90zw^{2}-2w^{3})}$ |
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 |
---|---|---|---|---|---|---|
12.18.0.i.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
30.18.0.a.1 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.18.1.j.1 | $60$ | $2$ | $2$ | $1$ | $0$ | 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.s.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.72.3.ci.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.72.3.ek.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.72.3.em.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.72.3.vr.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.72.3.vs.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.72.3.vz.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.72.3.wa.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.180.13.mu.1 | $60$ | $5$ | $5$ | $13$ | $6$ | $1^{12}$ |
60.216.13.ph.1 | $60$ | $6$ | $6$ | $13$ | $2$ | $1^{12}$ |
60.360.25.cbs.1 | $60$ | $10$ | $10$ | $25$ | $8$ | $1^{24}$ |
120.72.3.lk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.pd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bcl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bcz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.fdo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.fdv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ffs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.ffz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
180.108.5.bs.1 | $180$ | $3$ | $3$ | $5$ | $?$ | not computed |
180.324.21.bk.1 | $180$ | $9$ | $9$ | $21$ | $?$ | not computed |