Properties

Label 60.240.7-20.f.1.6
Level $60$
Index $240$
Genus $7$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $400$
Index: $240$ $\PSL_2$-index:$120$
Genus: $7 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $10^{4}\cdot20^{4}$ Cusp orbits $2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20B7
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.240.7.357

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}7&46\\16&35\end{bmatrix}$, $\begin{bmatrix}25&24\\6&25\end{bmatrix}$, $\begin{bmatrix}35&8\\42&55\end{bmatrix}$, $\begin{bmatrix}35&52\\2&21\end{bmatrix}$, $\begin{bmatrix}57&52\\34&43\end{bmatrix}$
Contains $-I$: no $\quad$ (see 20.120.7.f.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree: $48$
Cyclic 60-torsion field degree: $768$
Full 60-torsion field degree: $9216$

Jacobian

Conductor: $2^{20}\cdot5^{14}$
Simple: no
Squarefree: no
Decomposition: $1^{7}$
Newforms: 50.2.a.b$^{2}$, 100.2.a.a, 400.2.a.a, 400.2.a.b, 400.2.a.f, 400.2.a.g

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $ $=$ $ x t + x u - x v + z w $
$=$ $x w + x t - x u - y w + z w$
$=$ $y t + y u - y v + z w - 2 z u + z v$
$=$ $x z - 3 y z - z^{2} + t u + u^{2} - u v$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 25 x^{12} + 350 x^{10} y^{2} + 1225 x^{8} y^{4} - 300 x^{8} y^{2} z^{2} - 10 x^{8} z^{4} + \cdots + y^{4} z^{8} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 10.60.3.a.1 :

$\displaystyle X$ $=$ $\displaystyle -2x+y-3z$
$\displaystyle Y$ $=$ $\displaystyle 4x-2y+z$
$\displaystyle Z$ $=$ $\displaystyle x+2y-z$

Equation of the image curve:

$0$ $=$ $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 20.120.7.f.1 :

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

Equation of the image curve:

$0$ $=$ $ 25X^{12}+350X^{10}Y^{2}+1225X^{8}Y^{4}-300X^{8}Y^{2}Z^{2}-10X^{8}Z^{4}-2600X^{6}Y^{4}Z^{2}-145X^{6}Y^{2}Z^{4}+550X^{4}Y^{4}Z^{4}+50X^{4}Y^{2}Z^{6}+X^{4}Z^{8}-40X^{2}Y^{4}Z^{6}-7X^{2}Y^{2}Z^{8}+Y^{4}Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
60.120.3-10.a.1.2 $60$ $2$ $2$ $3$ $0$ $1^{4}$
60.120.3-10.a.1.5 $60$ $2$ $2$ $3$ $0$ $1^{4}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.480.13-20.e.1.3 $60$ $2$ $2$ $13$ $2$ $1^{6}$
60.480.13-20.h.1.3 $60$ $2$ $2$ $13$ $3$ $1^{6}$
60.480.13-20.i.1.4 $60$ $2$ $2$ $13$ $3$ $1^{6}$
60.480.13-20.l.1.6 $60$ $2$ $2$ $13$ $1$ $1^{6}$
60.480.13-60.u.1.6 $60$ $2$ $2$ $13$ $2$ $1^{6}$
60.480.13-60.x.1.8 $60$ $2$ $2$ $13$ $2$ $1^{6}$
60.480.13-60.bg.1.8 $60$ $2$ $2$ $13$ $3$ $1^{6}$
60.480.13-60.bj.1.8 $60$ $2$ $2$ $13$ $4$ $1^{6}$
60.480.15-20.g.1.2 $60$ $2$ $2$ $15$ $4$ $1^{8}$
60.480.15-20.h.1.2 $60$ $2$ $2$ $15$ $2$ $1^{8}$
60.480.15-20.h.1.4 $60$ $2$ $2$ $15$ $2$ $1^{8}$
60.480.15-20.m.1.4 $60$ $2$ $2$ $15$ $3$ $1^{8}$
60.480.15-20.m.1.8 $60$ $2$ $2$ $15$ $3$ $1^{8}$
60.480.15-20.n.1.4 $60$ $2$ $2$ $15$ $2$ $1^{8}$
60.480.15-20.n.1.8 $60$ $2$ $2$ $15$ $2$ $1^{8}$
60.480.15-60.o.1.4 $60$ $2$ $2$ $15$ $6$ $1^{8}$
60.480.15-60.o.1.16 $60$ $2$ $2$ $15$ $6$ $1^{8}$
60.480.15-60.q.1.4 $60$ $2$ $2$ $15$ $3$ $1^{8}$
60.480.15-60.q.1.16 $60$ $2$ $2$ $15$ $3$ $1^{8}$
60.480.15-60.bd.1.6 $60$ $2$ $2$ $15$ $5$ $1^{8}$
60.480.15-60.bd.1.16 $60$ $2$ $2$ $15$ $5$ $1^{8}$
60.480.15-60.be.1.4 $60$ $2$ $2$ $15$ $3$ $1^{8}$
60.480.15-60.be.1.16 $60$ $2$ $2$ $15$ $3$ $1^{8}$
60.720.19-20.t.1.1 $60$ $3$ $3$ $19$ $3$ $1^{12}$
60.720.27-60.dr.1.27 $60$ $3$ $3$ $27$ $10$ $1^{18}\cdot2$
60.960.33-60.x.1.4 $60$ $4$ $4$ $33$ $6$ $1^{26}$
120.480.13-40.o.1.2 $120$ $2$ $2$ $13$ $?$ not computed
120.480.13-40.x.1.2 $120$ $2$ $2$ $13$ $?$ not computed
120.480.13-40.ba.1.2 $120$ $2$ $2$ $13$ $?$ not computed
120.480.13-40.bj.1.2 $120$ $2$ $2$ $13$ $?$ not computed
120.480.13-120.ck.1.8 $120$ $2$ $2$ $13$ $?$ not computed
120.480.13-120.ct.1.12 $120$ $2$ $2$ $13$ $?$ not computed
120.480.13-120.du.1.8 $120$ $2$ $2$ $13$ $?$ not computed
120.480.13-120.ed.1.8 $120$ $2$ $2$ $13$ $?$ not computed
120.480.15-40.bd.1.5 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-40.bd.1.9 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-40.bg.1.5 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-40.bg.1.9 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-40.bs.1.5 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-40.bs.1.9 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-40.bv.1.5 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-40.bv.1.9 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-120.cc.1.7 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-120.cc.1.27 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-120.ci.1.11 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-120.ci.1.23 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-120.ds.1.13 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-120.ds.1.23 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-120.dv.1.7 $120$ $2$ $2$ $15$ $?$ not computed
120.480.15-120.dv.1.29 $120$ $2$ $2$ $15$ $?$ not computed