Invariants
Level: | $20$ | $\SL_2$-level: | $10$ | 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 | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{4}\cdot4$ | ||
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: | 10K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.72.1.20 |
Level structure
$\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}15&17\\14&17\end{bmatrix}$, $\begin{bmatrix}19&14\\10&17\end{bmatrix}$, $\begin{bmatrix}19&15\\4&7\end{bmatrix}$ |
$\GL_2(\Z/20\Z)$-subgroup: | $(C_2^3\times F_5):C_4$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 20-isogeny field degree: | $2$ |
Cyclic 20-torsion field degree: | $16$ |
Full 20-torsion field degree: | $640$ |
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 $ | $=$ | $ 5 x y - x z - y z $ |
$=$ | $5 x^{2} - 6 x z + 5 y^{2} - 6 y z + 4 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 8 x^{3} z + 5 x^{2} y^{2} + 28 x^{2} z^{2} - 10 x y^{2} z - 40 x z^{3} + 5 y^{2} z^{2} + 20 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{7562391552xz^{17}-23384125440xz^{15}w^{2}-188772480000xz^{13}w^{4}-155591744000xz^{11}w^{6}+217368800000xz^{9}w^{8}+210391200000xz^{7}w^{10}-19450250000xz^{5}w^{12}+537500000xz^{3}w^{14}-4687500xzw^{16}+7562391552yz^{17}-23384125440yz^{15}w^{2}-188772480000yz^{13}w^{4}-155591744000yz^{11}w^{6}+217368800000yz^{9}w^{8}+210391200000yz^{7}w^{10}-19450250000yz^{5}w^{12}+537500000yz^{3}w^{14}-4687500yzw^{16}-8453984256z^{18}-6208856064z^{16}w^{2}+134269977600z^{14}w^{4}+246801222400z^{12}w^{6}+13956992000z^{10}w^{8}-141092400000z^{8}w^{10}-28751150000z^{6}w^{12}+2322500000z^{4}w^{14}-53437500z^{2}w^{16}+390625w^{18}}{z^{2}(4376384xz^{15}+28964800xz^{13}w^{2}+73826500xz^{11}w^{4}+95231250xz^{9}w^{6}+69000000xz^{7}w^{8}+29537500xz^{5}w^{10}+7187500xz^{3}w^{12}+781250xzw^{14}+4376384yz^{15}+28964800yz^{13}w^{2}+73826500yz^{11}w^{4}+95231250yz^{9}w^{6}+69000000yz^{7}w^{8}+29537500yz^{5}w^{10}+7187500yz^{3}w^{12}+781250yzw^{14}-4892352z^{16}-29678928z^{14}w^{2}-69939060z^{12}w^{4}-83488775z^{10}w^{6}-56400625z^{8}w^{8}-23588750z^{6}w^{10}-6368750z^{4}w^{12}-1046875z^{2}w^{14}-78125w^{16})}$ |
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 |
---|---|---|---|---|---|---|
10.36.0.b.1 | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
20.24.1.f.2 | $20$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
20.36.0.a.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
20.36.1.d.1 | $20$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
20.144.5.t.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
20.144.5.x.1 | $20$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
20.144.5.bb.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
20.144.5.bf.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
20.360.13.u.1 | $20$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
40.144.5.fh.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.gj.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.io.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.jq.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.ih.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.144.5.ip.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.144.5.or.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
60.144.5.oz.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.216.13.fv.2 | $60$ | $3$ | $3$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
60.288.13.nh.2 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
100.360.13.j.1 | $100$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.144.5.cnl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cpp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eeg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
140.144.5.dx.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
140.144.5.dz.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
140.144.5.fd.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
140.144.5.ff.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
220.144.5.dx.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
220.144.5.dz.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
220.144.5.fd.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
220.144.5.ff.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
260.144.5.dx.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
260.144.5.dz.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
260.144.5.fd.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
260.144.5.ff.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bez.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bfn.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bno.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.boc.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |