Invariants
Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $900$ | ||
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: | $1$ | ||||||
$\Q$-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: | 60.72.1.424 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&10\\29&19\end{bmatrix}$, $\begin{bmatrix}43&55\\13&26\end{bmatrix}$, $\begin{bmatrix}44&5\\47&43\end{bmatrix}$, $\begin{bmatrix}49&30\\33&37\end{bmatrix}$ |
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: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 900.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ - 5 x w + z^{2} $ |
$=$ | $5 x^{2} + 2 x w - 3 y^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 2 x^{2} z^{2} - 15 y^{2} z^{2} + 5 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{1708593750xy^{16}w+6264843750xy^{14}w^{3}+21740906250xy^{12}w^{5}+50177981250xy^{10}w^{7}+31569243750xy^{8}w^{9}+32664890250xy^{6}w^{11}+2447671500xy^{4}w^{13}-253083840xy^{2}w^{15}-4376384xw^{17}-284765625y^{18}-1879453125y^{16}w^{2}-11390625000y^{14}w^{4}-14144878125y^{12}w^{6}-24959137500y^{10}w^{8}-25000214625y^{8}w^{10}+11568316725y^{6}w^{12}-1568527200y^{4}w^{14}-133433904y^{2}w^{16}+515968w^{18}}{w^{3}(170859375xy^{14}+877078125xy^{12}w^{2}-621928125xy^{10}w^{4}+33665625xy^{8}w^{6}+65188125xy^{6}w^{8}-19269225xy^{4}w^{10}+2021745xy^{2}w^{12}-68381xw^{14}-478406250y^{14}w+95681250y^{12}w^{3}+312558750y^{10}w^{5}-215763750y^{8}w^{7}+57233250y^{6}w^{9}-6634170y^{4}w^{11}+222954y^{2}w^{13}+8062w^{15})}$ |
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 |
---|---|---|---|---|---|---|
30.36.1.p.1 | $30$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
60.36.0.j.2 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.0.ch.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 |
---|---|---|---|---|---|---|
60.144.9.bj.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.bm.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.fa.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.fc.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.fy.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.gc.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.ih.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.il.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.216.9.bb.1 | $60$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.288.13.sh.1 | $60$ | $4$ | $4$ | $13$ | $5$ | $1^{6}\cdot2^{3}$ |
60.360.21.cx.1 | $60$ | $5$ | $5$ | $21$ | $6$ | $1^{8}\cdot2^{6}$ |
120.144.9.izk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.jaf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rbi.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rcd.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rvv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rwx.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.smm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.snv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
180.216.13.ik.2 | $180$ | $3$ | $3$ | $13$ | $?$ | not computed |
300.360.21.q.1 | $300$ | $5$ | $5$ | $21$ | $?$ | not computed |