Invariants
Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $80$ | ||
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 | $2^{4}\cdot4$ | ||
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: | 20H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.72.1.53 |
Level structure
$\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}7&11\\16&11\end{bmatrix}$, $\begin{bmatrix}9&9\\14&5\end{bmatrix}$, $\begin{bmatrix}9&15\\14&17\end{bmatrix}$ |
$\GL_2(\Z/20\Z)$-subgroup: | $C_{10}.C_4^3$ |
Contains $-I$: | yes |
Quadratic refinements: | 40.144.1-20.e.1.1, 40.144.1-20.e.1.2, 40.144.1-20.e.1.3, 40.144.1-20.e.1.4, 40.144.1-20.e.1.5, 40.144.1-20.e.1.6, 40.144.1-20.e.1.7, 40.144.1-20.e.1.8, 120.144.1-20.e.1.1, 120.144.1-20.e.1.2, 120.144.1-20.e.1.3, 120.144.1-20.e.1.4, 120.144.1-20.e.1.5, 120.144.1-20.e.1.6, 120.144.1-20.e.1.7, 120.144.1-20.e.1.8, 280.144.1-20.e.1.1, 280.144.1-20.e.1.2, 280.144.1-20.e.1.3, 280.144.1-20.e.1.4, 280.144.1-20.e.1.5, 280.144.1-20.e.1.6, 280.144.1-20.e.1.7, 280.144.1-20.e.1.8 |
Cyclic 20-isogeny field degree: | $2$ |
Cyclic 20-torsion field degree: | $16$ |
Full 20-torsion field degree: | $640$ |
Jacobian
Conductor: | $2^{4}\cdot5$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 80.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x z + 2 y^{2} + 3 y z - 2 z^{2} $ |
$=$ | $5 x^{2} + 10 x y + 5 y^{2} + 5 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} - 8 x^{3} z - 4 x^{2} z^{2} + 8 x z^{3} + 5 y^{2} z^{2} + 29 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{5^5}\cdot\frac{105468750xz^{17}+84375000xz^{15}w^{2}-28125000xz^{13}w^{4}-55250000xz^{11}w^{6}-26500000xz^{9}w^{8}-6480000xz^{7}w^{10}-884000xz^{5}w^{12}-64000xz^{3}w^{14}-1920xzw^{16}+105468750yz^{17}+84375000yz^{15}w^{2}-28125000yz^{13}w^{4}-55250000yz^{11}w^{6}-26500000yz^{9}w^{8}-6480000yz^{7}w^{10}-884000yz^{5}w^{12}-64000yz^{3}w^{14}-1920yzw^{16}-580078125z^{18}-1181250000z^{16}w^{2}-998437500z^{14}w^{4}-457312500z^{12}w^{6}-124200000z^{10}w^{8}-20460000z^{8}w^{10}-1974000z^{6}w^{12}-96000z^{4}w^{14}-960z^{2}w^{16}+64w^{18}}{z^{10}(5z^{2}+w^{2})^{2}(50xz^{3}+20xzw^{2}+50yz^{3}+20yzw^{2}-275z^{4}-75z^{2}w^{2}-4w^{4})}$ |
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 |
---|---|---|---|---|---|---|
20.36.0.a.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
20.36.0.b.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
20.36.1.b.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.c.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
20.144.5.j.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
20.144.5.r.2 | $20$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
20.144.5.t.2 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
20.360.13.h.1 | $20$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
40.144.5.z.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.co.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.el.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.fe.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.eb.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.144.5.ed.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.fp.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.144.5.fr.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
60.216.13.z.1 | $60$ | $3$ | $3$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
60.288.13.id.1 | $60$ | $4$ | $4$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
100.360.13.e.1 | $100$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.144.5.bci.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.bcw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.boi.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.bow.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
140.144.5.cg.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
140.144.5.ch.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
140.144.5.co.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
140.144.5.cp.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
220.144.5.cg.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
220.144.5.ch.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
220.144.5.co.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
220.144.5.cp.2 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
260.144.5.cg.2 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
260.144.5.ch.2 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
260.144.5.co.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
260.144.5.cp.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.pq.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.px.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.ru.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.sb.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |