Properties

Label 120.96.1-40.bc.1.16
Level $120$
Index $96$
Genus $1$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $120$ $\SL_2$-level: $8$ Newform level: $800$
Index: $96$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2 \le \gamma \le 48$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G1

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}19&36\\84&37\end{bmatrix}$, $\begin{bmatrix}35&66\\72&31\end{bmatrix}$, $\begin{bmatrix}39&116\\88&45\end{bmatrix}$, $\begin{bmatrix}61&38\\92&105\end{bmatrix}$, $\begin{bmatrix}79&6\\36&61\end{bmatrix}$
Contains $-I$: no $\quad$ (see 40.48.1.bc.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree: $48$
Cyclic 120-torsion field degree: $1536$
Full 120-torsion field degree: $368640$

Jacobian

Conductor: $?$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 800.2.a.d

Models

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

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

$ 0 $ $=$ $ x^{4} + 5 x^{2} y^{2} + 3 x^{2} z^{2} + 2 z^{4} $
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{5^3}\cdot\frac{1968750y^{2}z^{10}+4725000y^{2}z^{8}w^{2}+360000y^{2}z^{6}w^{4}-288000y^{2}z^{4}w^{6}-2419200y^{2}z^{2}w^{8}-645120y^{2}w^{10}+484375z^{12}+750000z^{10}w^{2}-480000z^{8}w^{4}-512000z^{6}w^{6}-1632000z^{4}w^{8}-983040z^{2}w^{10}-131072w^{12}}{w^{4}z^{4}(10y^{2}z^{2}-8y^{2}w^{2}+5z^{4})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.48.1.bc.1 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{5}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}z$

Equation of the image curve:

$0$ $=$ $ X^{4}+5X^{2}Y^{2}+3X^{2}Z^{2}+2Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.48.0-8.d.2.9 $24$ $2$ $2$ $0$ $0$ full Jacobian
120.48.0-8.d.2.6 $120$ $2$ $2$ $0$ $?$ full Jacobian
120.48.0-40.h.1.5 $120$ $2$ $2$ $0$ $?$ full Jacobian
120.48.0-40.h.1.19 $120$ $2$ $2$ $0$ $?$ full Jacobian
120.48.1-40.c.1.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1-40.c.1.17 $120$ $2$ $2$ $1$ $?$ 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
120.192.1-40.b.2.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-40.h.1.6 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-40.bb.1.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-40.bd.2.8 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-40.bt.1.5 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-40.bv.2.7 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-40.ca.2.5 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-40.cb.1.7 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.fq.2.16 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.fu.2.15 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.gx.2.7 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.hb.2.8 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.lz.2.7 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.md.2.12 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.nf.2.16 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.nj.2.15 $120$ $2$ $2$ $1$ $?$ dimension zero
120.288.9-120.ro.1.36 $120$ $3$ $3$ $9$ $?$ not computed
120.384.9-120.je.1.17 $120$ $4$ $4$ $9$ $?$ not computed
120.480.17-40.bw.1.5 $120$ $5$ $5$ $17$ $?$ not computed