Invariants
Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $4$ | ||||||
$\overline{\Q}$-gonality: | $4$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 20I5 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.288.5.964 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}7&30\\18&31\end{bmatrix}$, $\begin{bmatrix}17&5\\58&29\end{bmatrix}$, $\begin{bmatrix}27&40\\4&31\end{bmatrix}$, $\begin{bmatrix}33&40\\40&41\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 60.144.5.cq.1 for the level structure with $-I$) |
Cyclic 60-isogeny field degree: | $8$ |
Cyclic 60-torsion field degree: | $64$ |
Full 60-torsion field degree: | $7680$ |
Jacobian
Conductor: | $2^{17}\cdot3^{8}\cdot5^{7}$ |
Simple: | no |
Squarefree: | yes |
Decomposition: | $1^{3}\cdot2$ |
Newforms: | 80.2.a.b, 360.2.f.c, 1800.2.a.v, 3600.2.a.be |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
$ 0 $ | $=$ | $ y z - z^{2} + w^{2} $ |
$=$ | $y^{2} - 4 y w + t^{2}$ | |
$=$ | $15 x^{2} - y w - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 900 x^{4} y^{4} - 3600 x^{3} y^{5} + 3000 x^{2} y^{6} - 1080 x^{2} y^{4} z^{2} - 480 x^{2} y^{2} z^{4} + \cdots + 49 z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=19$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{126976yw^{17}-1232896yw^{15}t^{2}+4208640yw^{13}t^{4}-5876224yw^{11}t^{6}+2836480yw^{9}t^{8}-598272yw^{7}t^{10}+57392yw^{5}t^{12}-2240yw^{3}t^{14}+24ywt^{16}+4096w^{18}-28672w^{16}t^{2}+307200w^{14}t^{4}-1032960w^{12}t^{6}+1405696w^{10}t^{8}-625152w^{8}t^{10}+115504w^{6}t^{12}-8960w^{4}t^{14}+240w^{2}t^{16}-t^{18}}{t^{2}w^{10}(1600yw^{5}-360yw^{3}t^{2}+14ywt^{4}-400w^{4}t^{2}+65w^{2}t^{4}-t^{6})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 60.144.5.cq.1 :
$\displaystyle X$ | $=$ | $\displaystyle x+z$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
$0$ | $=$ | $ 900X^{4}Y^{4}-3600X^{3}Y^{5}+3000X^{2}Y^{6}-1080X^{2}Y^{4}Z^{2}-480X^{2}Y^{2}Z^{4}+1200XY^{7}+2160XY^{5}Z^{2}+960XY^{3}Z^{4}-275Y^{8}-240Y^{6}Z^{2}+154Y^{4}Z^{4}+168Y^{2}Z^{6}+49Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
20.144.1-20.b.1.3 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
60.144.1-20.b.1.12 | $60$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
60.144.1-60.ch.2.8 | $60$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
60.144.1-60.ch.2.13 | $60$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
60.144.3-60.xu.2.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
60.144.3-60.xu.2.15 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
60.576.13-60.gi.2.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{2}\cdot4$ |
60.576.13-60.gj.2.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{2}\cdot4$ |
60.576.13-60.gk.1.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{2}\cdot4$ |
60.576.13-60.gl.1.2 | $60$ | $2$ | $2$ | $13$ | $1$ | $2^{2}\cdot4$ |
60.864.29-60.ku.2.17 | $60$ | $3$ | $3$ | $29$ | $3$ | $1^{12}\cdot2^{6}$ |
60.1152.33-60.cy.2.11 | $60$ | $4$ | $4$ | $33$ | $7$ | $1^{14}\cdot2^{7}$ |
60.1440.37-60.co.1.2 | $60$ | $5$ | $5$ | $37$ | $8$ | $1^{16}\cdot2^{8}$ |