Invariants
Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
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: | 20H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.388 |
Level structure
$\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}13&45\\0&19\end{bmatrix}$, $\begin{bmatrix}17&5\\58&53\end{bmatrix}$, $\begin{bmatrix}39&35\\4&39\end{bmatrix}$, $\begin{bmatrix}49&50\\32&27\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 60-isogeny field degree: | $8$ |
Cyclic 60-torsion field degree: | $128$ |
Full 60-torsion field degree: | $30720$ |
Jacobian
Conductor: | $2^{4}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 400.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 x y - 2 x z + 2 y^{2} - y z + 2 z^{2} $ |
$=$ | $4 x^{2} + 3 x y - 3 x z - 2 y^{2} + y z - 2 z^{2} - 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 25 x^{4} - 15 x^{3} y + 3 x^{2} y^{2} - 30 x^{2} z^{2} + 9 x y z^{2} + 18 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle 5z$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{3}{2^4}\cdot\frac{98947265625xz^{17}-412829296875xz^{15}w^{2}+66791250000xz^{13}w^{4}+422184000000xz^{11}w^{6}-286815000000xz^{9}w^{8}-81056160000xz^{7}w^{10}+64678080000xz^{5}w^{12}+14633779200xz^{3}w^{14}+275374080xzw^{16}-79880859375y^{2}z^{16}+521160937500y^{2}z^{14}w^{2}-363939375000y^{2}z^{12}w^{4}-154692000000y^{2}z^{10}w^{6}+252831000000y^{2}z^{8}w^{8}-55372800000y^{2}z^{6}w^{10}-20058816000y^{2}z^{4}w^{12}-2871705600y^{2}z^{2}w^{14}-64696320y^{2}w^{16}+19066406250yz^{17}-218657812500yz^{15}w^{2}+90947812500yz^{13}w^{4}+66297000000yz^{11}w^{6}-25704000000yz^{9}w^{8}-39159360000yz^{7}w^{10}+19760544000yz^{5}w^{12}+4619980800yz^{3}w^{14}-206807040yzw^{16}-50828125000z^{18}+587211328125z^{16}w^{2}-179066718750z^{14}w^{4}-612535625000z^{12}w^{6}+262371150000z^{10}w^{8}+260183040000z^{8}w^{10}-89142304000z^{6}w^{12}-45420249600z^{4}w^{14}-3315271680z^{2}w^{16}-26099712w^{18}}{w^{4}(38671875xz^{13}+60234375xz^{11}w^{2}+40500000xz^{9}w^{4}+16440000xz^{7}w^{6}+4536000xz^{5}w^{8}+758400xz^{3}w^{10}+53760xzw^{12}-20703125y^{2}z^{12}-47937500y^{2}z^{10}w^{2}-41125000y^{2}z^{8}w^{4}-17920000y^{2}z^{6}w^{6}-3912000y^{2}z^{4}w^{8}-243200y^{2}z^{2}w^{10}+12800y^{2}w^{12}+17968750yz^{13}+4562500yz^{11}w^{2}-12062500yz^{9}w^{4}-9520000yz^{7}w^{6}-2688000yz^{5}w^{8}-166400yz^{3}w^{10}+47360yzw^{12}-59375000z^{14}-123640625z^{12}w^{2}-106656250z^{10}w^{4}-52975000z^{8}w^{6}-16854000z^{6}w^{8}-3267200z^{4}w^{10}-286720z^{2}w^{12}+1536w^{14})}$ |
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 |
---|---|---|---|---|---|---|
20.36.1.e.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
30.36.0.b.2 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
60.36.0.d.2 | $60$ | $2$ | $2$ | $0$ | $0$ | 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 |
---|---|---|---|---|---|---|
60.144.5.bo.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.co.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.144.5.ht.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.144.5.hw.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.144.5.pi.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.pn.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.ps.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.144.5.pu.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.216.13.ga.1 | $60$ | $3$ | $3$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
60.288.13.nm.2 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
60.360.13.by.1 | $60$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
120.144.5.ii.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.rx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cjq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ckl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eiu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ekd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.elo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eme.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
300.360.13.y.2 | $300$ | $5$ | $5$ | $13$ | $?$ | not computed |