Invariants
Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $3600$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot30^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ 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: | 30C5 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.5.77 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&55\\23&8\end{bmatrix}$, $\begin{bmatrix}21&16\\49&3\end{bmatrix}$, $\begin{bmatrix}43&0\\21&37\end{bmatrix}$, $\begin{bmatrix}54&55\\17&27\end{bmatrix}$ |
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: | $2^{20}\cdot3^{8}\cdot5^{8}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{3}\cdot2$ |
Newforms: | 400.2.a.c, 720.2.f.d, 3600.2.a.e$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{7}$
$ 0 $ | $=$ | $ t r + u^{2} $ |
$=$ | $ - w r + t u$ | |
$=$ | $y r + t^{2}$ | |
$=$ | $ - y r + w u$ | |
$=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 125 x^{12} + 22 x^{6} z^{6} + 5 y^{2} z^{10} + z^{12} $ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ -5x^{12} - 110x^{6} - 625 $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle u$ |
$\displaystyle Y$ | $=$ | $\displaystyle v$ |
$\displaystyle Z$ | $=$ | $\displaystyle r$ |
Birational map from embedded model to Weierstrass model:
$\displaystyle X$ | $=$ | $\displaystyle r$ |
$\displaystyle Y$ | $=$ | $\displaystyle -5vr^{5}$ |
$\displaystyle Z$ | $=$ | $\displaystyle -u$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{267187500zv^{4}r-7828380000zv^{2}r^{3}+118162368zr^{5}-1953125v^{6}+515475000v^{4}r^{2}-1073051280v^{2}r^{4}+8895744r^{6}}{r(390625zv^{4}-751250zv^{2}r^{2}+68381zr^{4}-137500v^{4}r-1760v^{2}r^{3}+5148r^{5})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(3)$ | $3$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
20.24.1.f.1 | $20$ | $3$ | $3$ | $1$ | $0$ | $1^{2}\cdot2$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
20.24.1.f.1 | $20$ | $3$ | $3$ | $1$ | $0$ | $1^{2}\cdot2$ |
30.36.0.f.2 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.2.fs.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
60.36.3.c.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
60.144.9.gd.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
60.144.9.ge.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
60.144.9.hf.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
60.144.9.hg.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
60.144.9.it.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
60.144.9.iu.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
60.144.9.jy.2 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
60.144.9.jz.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
60.216.13.fv.1 | $60$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{2}$ |
60.288.21.gz.1 | $60$ | $4$ | $4$ | $21$ | $2$ | $1^{8}\cdot2^{4}$ |
60.360.25.zf.1 | $60$ | $5$ | $5$ | $25$ | $2$ | $1^{8}\cdot2^{6}$ |
120.144.9.rwz.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rxg.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.sfp.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.sfw.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tbc.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tbj.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tke.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.tkl.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
180.216.17.cq.1 | $180$ | $3$ | $3$ | $17$ | $?$ | not computed |