Invariants
Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $80$ | ||
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: | 60.72.1.272 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&55\\55&42\end{bmatrix}$, $\begin{bmatrix}19&5\\49&4\end{bmatrix}$, $\begin{bmatrix}49&50\\19&43\end{bmatrix}$, $\begin{bmatrix}51&5\\1&42\end{bmatrix}$ |
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: | $2^{4}\cdot5$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 y^{2} + y z + 2 z^{2} - 2 z w + w^{2} $ |
$=$ | $15 x^{2} + y z - y w + z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + 5 x^{3} z - 51 x^{2} y^{2} + 6 x^{2} z^{2} - 24 x y^{2} z + 2 x z^{3} + 180 y^{4} - 24 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^{30}}\cdot\frac{1131753626219yz^{17}-58968692067448yz^{16}w+354826487055824yz^{15}w^{2}-372710010639520yz^{14}w^{3}-1648907491712080yz^{13}w^{4}+4770044482656128yz^{12}w^{5}-5080414398528256yz^{11}w^{6}+2513566719392768yz^{10}w^{7}-218967994812160yz^{9}w^{8}+47968122664960yz^{8}w^{9}-363887215935488yz^{7}w^{10}+201462973382656yz^{6}w^{11}+177847134625792yz^{5}w^{12}-219035139768320yz^{4}w^{13}+96968780677120yz^{3}w^{14}-7397044649984yz^{2}w^{15}-5431293181952yzw^{16}+2056497135616yw^{17}+3275981939334z^{18}-26408214047654z^{17}w-74592337587749z^{16}w^{2}+998294866236664z^{15}w^{3}-2783355210356900z^{14}w^{4}+3115884086876128z^{13}w^{5}-917441127870128z^{12}w^{6}-1143940833605888z^{11}w^{7}+687724582227328z^{10}w^{8}+1084096583718400z^{9}w^{9}-1699363565809408z^{8}w^{10}+1041265394413568z^{7}w^{11}-260367488714752z^{6}w^{12}-9602687746048z^{5}w^{13}+24327965696000z^{4}w^{14}-7269044322304z^{3}w^{15}+10831036350464z^{2}w^{16}-6483497844736zw^{17}+2140672557056w^{18}}{(z-w)^{15}(29yz^{2}+92yzw+44yw^{2}-6z^{3}+46z^{2}w+61zw^{2}+4w^{3})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
15.36.0.b.1 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.0.cg.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.1.ga.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 |
---|---|---|---|---|---|---|
60.144.9.bd.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.be.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.cp.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.cq.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.iw.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.iy.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.144.9.jd.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.jf.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.216.9.bh.2 | $60$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
60.288.13.sk.1 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
60.360.21.da.1 | $60$ | $5$ | $5$ | $21$ | $6$ | $1^{8}\cdot2^{6}$ |
120.144.9.ixn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.ixu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.jtv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.juc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tcd.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tcy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.teh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tfc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
180.216.13.ho.1 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
300.360.21.s.1 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |