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Properties

Label 20.60.4.c.1
Level $20$
Index $60$
Genus $4$
Analytic rank $1$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $20$ $\SL_2$-level: $20$ Newform level: $400$
Index: $60$ $\PSL_2$-index:$60$
Genus: $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $10^{2}\cdot20^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3 \le \gamma \le 4$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20A4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 20.60.4.11

Level structure

$\GL_2(\Z/20\Z)$-generators: $\begin{bmatrix}9&0\\10&7\end{bmatrix}$, $\begin{bmatrix}11&4\\4&9\end{bmatrix}$, $\begin{bmatrix}13&2\\14&19\end{bmatrix}$, $\begin{bmatrix}13&8\\6&9\end{bmatrix}$
$\GL_2(\Z/20\Z)$-subgroup: $C_2^3\times \GU(2,3)$
Contains $-I$: yes
Quadratic refinements: 20.120.4-20.c.1.1, 20.120.4-20.c.1.2, 20.120.4-20.c.1.3, 20.120.4-20.c.1.4, 40.120.4-20.c.1.1, 40.120.4-20.c.1.2, 40.120.4-20.c.1.3, 40.120.4-20.c.1.4, 60.120.4-20.c.1.1, 60.120.4-20.c.1.2, 60.120.4-20.c.1.3, 60.120.4-20.c.1.4, 120.120.4-20.c.1.1, 120.120.4-20.c.1.2, 120.120.4-20.c.1.3, 120.120.4-20.c.1.4, 140.120.4-20.c.1.1, 140.120.4-20.c.1.2, 140.120.4-20.c.1.3, 140.120.4-20.c.1.4, 220.120.4-20.c.1.1, 220.120.4-20.c.1.2, 220.120.4-20.c.1.3, 220.120.4-20.c.1.4, 260.120.4-20.c.1.1, 260.120.4-20.c.1.2, 260.120.4-20.c.1.3, 260.120.4-20.c.1.4, 280.120.4-20.c.1.1, 280.120.4-20.c.1.2, 280.120.4-20.c.1.3, 280.120.4-20.c.1.4
Cyclic 20-isogeny field degree: $12$
Cyclic 20-torsion field degree: $96$
Full 20-torsion field degree: $768$

Jacobian

Conductor: $2^{10}\cdot5^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{4}$
Newforms: 50.2.a.b$^{2}$, 400.2.a.d, 400.2.a.h

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 10 x^{2} + 5 x y + 5 y^{2} - z w + w^{2} $
$=$ $5 x y^{2} - x z^{2} + x z w - 5 y^{3} - y z^{2} + 3 y z w - y w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 500 x^{6} - 125 x^{4} y^{2} + 25 x^{4} y z + 150 x^{4} z^{2} + 10 x^{2} y^{4} - 15 x^{2} y^{3} z + \cdots + y^{2} z^{4} $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:1:0)$, $(0:0:1:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x+y$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{28225xyz^{8}-81685xyz^{7}w-122880xyz^{6}w^{2}+807050xyz^{5}w^{3}-964575xyz^{4}w^{4}-381145xyz^{3}w^{5}+1441890xyz^{2}w^{6}-895180xyzw^{7}+162760xyw^{8}+5715y^{2}z^{8}-38455y^{2}z^{7}w+16520y^{2}z^{6}w^{2}+430110y^{2}z^{5}w^{3}-1108225y^{2}z^{4}w^{4}+815635y^{2}z^{3}w^{5}+302570y^{2}z^{2}w^{6}-569660y^{2}zw^{7}+162760y^{2}w^{8}-2048z^{10}+12491z^{9}w-29060z^{8}w^{2}+25933z^{7}w^{3}+2390z^{6}w^{4}+7291z^{5}w^{5}-89468z^{4}w^{6}+131925z^{3}w^{7}-78642z^{2}w^{8}+17356zw^{9}-216w^{10}}{85xyz^{8}-375xyz^{7}w+575xyz^{6}w^{2}-555xyz^{5}w^{3}+325xyz^{4}w^{4}-5xyz^{3}w^{5}-115xyz^{2}w^{6}+55xyzw^{7}-10xyw^{8}+15y^{2}z^{8}-125y^{2}z^{7}w+165y^{2}z^{6}w^{2}-25y^{2}z^{5}w^{3}-25y^{2}z^{4}w^{4}+65y^{2}z^{3}w^{5}-45y^{2}z^{2}w^{6}+35y^{2}zw^{7}-10y^{2}w^{8}+7z^{9}w-30z^{8}w^{2}+60z^{7}w^{3}-68z^{6}w^{4}+34z^{5}w^{5}+8z^{4}w^{6}-10z^{3}w^{7}-8z^{2}w^{8}+9zw^{9}-2w^{10}}$

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
10.30.2.a.1 $10$ $2$ $2$ $2$ $0$ $1^{2}$
20.12.0.a.1 $20$ $5$ $5$ $0$ $0$ full Jacobian
20.30.2.f.1 $20$ $2$ $2$ $2$ $1$ $1^{2}$
20.30.2.j.1 $20$ $2$ $2$ $2$ $0$ $1^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
20.120.8.c.1 $20$ $2$ $2$ $8$ $3$ $1^{4}$
20.120.8.e.1 $20$ $2$ $2$ $8$ $1$ $1^{4}$
20.180.10.c.1 $20$ $3$ $3$ $10$ $2$ $1^{6}$
20.240.13.m.1 $20$ $4$ $4$ $13$ $2$ $1^{9}$
40.120.8.e.1 $40$ $2$ $2$ $8$ $3$ $1^{4}$
40.120.8.i.1 $40$ $2$ $2$ $8$ $3$ $1^{4}$
60.120.8.g.1 $60$ $2$ $2$ $8$ $3$ $1^{4}$
60.120.8.i.1 $60$ $2$ $2$ $8$ $1$ $1^{4}$
60.180.14.e.1 $60$ $3$ $3$ $14$ $6$ $1^{10}$
60.240.17.e.1 $60$ $4$ $4$ $17$ $4$ $1^{13}$
120.120.8.p.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.v.1 $120$ $2$ $2$ $8$ $?$ not computed
140.120.8.g.1 $140$ $2$ $2$ $8$ $?$ not computed
140.120.8.h.1 $140$ $2$ $2$ $8$ $?$ not computed
220.120.8.g.1 $220$ $2$ $2$ $8$ $?$ not computed
220.120.8.h.1 $220$ $2$ $2$ $8$ $?$ not computed
260.120.8.g.1 $260$ $2$ $2$ $8$ $?$ not computed
260.120.8.h.1 $260$ $2$ $2$ $8$ $?$ not computed
280.120.8.o.1 $280$ $2$ $2$ $8$ $?$ not computed
280.120.8.r.1 $280$ $2$ $2$ $8$ $?$ not computed