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Properties

Label 60.40.1.b.1
Level $60$
Index $40$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $10$ Newform level: $400$
Index: $40$ $\PSL_2$-index:$40$
Genus: $1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $10^{4}$ Cusp orbits $4$
Elliptic points: $0$ of order $2$ and $4$ 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: 10H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.40.1.24

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}9&26\\1&17\end{bmatrix}$, $\begin{bmatrix}23&33\\8&17\end{bmatrix}$, $\begin{bmatrix}34&55\\7&21\end{bmatrix}$, $\begin{bmatrix}50&17\\57&56\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.80.1-60.b.1.1, 60.80.1-60.b.1.2, 60.80.1-60.b.1.3, 60.80.1-60.b.1.4, 120.80.1-60.b.1.1, 120.80.1-60.b.1.2, 120.80.1-60.b.1.3, 120.80.1-60.b.1.4
Cyclic 60-isogeny field degree: $144$
Cyclic 60-torsion field degree: $2304$
Full 60-torsion field degree: $55296$

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

$ 0 $ $=$ $ x^{4} - 5 x^{3} z - 9 x^{2} y^{2} + 10 x z^{3} - 36 y^{4} - 5 z^{4} $
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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 between models of this curve

Birational map from embedded model to plane model:

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

Maps to other modular curves

$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 3\cdot5^2\,\frac{8470288yz^{9}+10149600yz^{8}w+21969096yz^{7}w^{2}+10982400yz^{6}w^{3}+13925229yz^{5}w^{4}+972390yz^{4}w^{5}+3355722yz^{3}w^{6}-877740yz^{2}w^{7}+489840yzw^{8}-80600yw^{9}+5701472z^{10}+12525216z^{9}w+26517984z^{8}w^{2}+25888752z^{7}w^{3}+26180646z^{6}w^{4}+11840178z^{5}w^{5}+7963203z^{4}w^{6}+706224z^{3}w^{7}+1013595z^{2}w^{8}-252520zw^{9}+61500w^{10}}{11810yz^{9}-63765yz^{8}w+158850yz^{7}w^{2}-246990yz^{6}w^{3}+277065yz^{5}w^{4}-204345yz^{4}w^{5}+92130yz^{3}w^{6}-24075yz^{2}w^{7}+3330yzw^{8}-205yw^{9}+7873z^{10}-34650z^{9}w+70260z^{8}w^{2}-101640z^{7}w^{3}+121560z^{6}w^{4}-142062z^{5}w^{5}+116325z^{4}w^{6}-56445z^{3}w^{7}+15855z^{2}w^{8}-2420zw^{9}+132w^{10}}$

Modular covers

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Cover information

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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
4.2.0.a.1 $4$ $20$ $20$ $0$ $0$ full Jacobian
15.20.0.a.1 $15$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
15.20.0.a.1 $15$ $2$ $2$ $0$ $0$ full Jacobian
20.20.1.a.1 $20$ $2$ $2$ $1$ $0$ dimension zero
60.20.0.d.1 $60$ $2$ $2$ $0$ $0$ 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
60.120.5.c.1 $60$ $3$ $3$ $5$ $0$ $1^{4}$
60.120.5.s.1 $60$ $3$ $3$ $5$ $1$ $1^{4}$
60.120.7.bg.1 $60$ $3$ $3$ $7$ $1$ $1^{4}\cdot2$
60.120.9.k.1 $60$ $3$ $3$ $9$ $2$ $1^{6}\cdot2$
60.160.9.e.1 $60$ $4$ $4$ $9$ $2$ $1^{8}$
60.160.9.o.1 $60$ $4$ $4$ $9$ $5$ $1^{8}$
300.200.9.e.1 $300$ $5$ $5$ $9$ $?$ not computed