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Properties

Label 120.144.3-15.a.1.10
Level $120$
Index $144$
Genus $3$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $120$ $\SL_2$-level: $30$ Newform level: $45$
Index: $144$ $\PSL_2$-index:$72$
Genus: $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $3^{4}\cdot15^{4}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $3$
$\overline{\Q}$-gonality: $3$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 15E3

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}41&45\\99&62\end{bmatrix}$, $\begin{bmatrix}75&97\\79&108\end{bmatrix}$, $\begin{bmatrix}95&102\\3&29\end{bmatrix}$, $\begin{bmatrix}102&31\\19&84\end{bmatrix}$, $\begin{bmatrix}104&117\\63&23\end{bmatrix}$, $\begin{bmatrix}117&26\\35&63\end{bmatrix}$
Contains $-I$: no $\quad$ (see 15.72.3.a.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree: $12$
Cyclic 120-torsion field degree: $384$
Full 120-torsion field degree: $245760$

Models

Canonical model in $\mathbb{P}^{ 2 }$

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

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(1:0:0)$, $(-2:-1:1)$, $(1:2:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{x^{18}-714x^{16}yz-174951x^{15}z^{3}+15875374x^{13}yz^{4}+294242376x^{12}z^{6}-2231220361x^{10}yz^{7}-6838076244x^{9}z^{9}+9461848986x^{7}yz^{10}-31494996496x^{6}z^{12}+14122661253x^{4}yz^{13}+23950271597x^{3}z^{15}-93750xy^{16}z+1468750xy^{13}z^{4}+1890625xy^{10}z^{7}+19940202xy^{7}z^{10}-3299371917xy^{4}z^{13}+11703100608xyz^{16}+15625y^{18}-140625y^{15}z^{3}-375000y^{12}z^{6}-55952412y^{9}z^{9}+2642480720y^{6}z^{12}-29288460677y^{3}z^{15}+86629783010z^{18}}{z(x^{16}y-30x^{15}z^{2}-391x^{13}yz^{3}+2741x^{12}z^{5}+11197x^{10}yz^{6}-8654x^{9}z^{8}+82705x^{7}yz^{9}-542666x^{6}z^{11}-1177994x^{4}yz^{12}-4357560x^{3}z^{14}-15625xy^{10}z^{6}-15204xy^{7}z^{9}+1538129xy^{4}z^{12}-5301770xyz^{15}+93749y^{9}z^{8}-1092023y^{6}z^{11}+5501185y^{3}z^{14}-11104775z^{17})}$

Modular covers

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

Factor curve Level Index Degree Genus Rank
$X_0(5)$ $5$ $24$ $12$ $0$ $0$
24.24.0-3.a.1.4 $24$ $6$ $6$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
24.24.0-3.a.1.4 $24$ $6$ $6$ $0$ $0$
120.48.1-15.a.1.18 $120$ $3$ $3$ $1$ $?$
120.48.1-15.a.1.20 $120$ $3$ $3$ $1$ $?$

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

Covering curve Level Index Degree Genus
120.288.5-15.a.1.14 $120$ $2$ $2$ $5$
120.288.5-15.a.2.16 $120$ $2$ $2$ $5$
120.288.5-15.b.1.8 $120$ $2$ $2$ $5$
120.288.5-15.b.2.6 $120$ $2$ $2$ $5$
120.288.5-60.ke.1.2 $120$ $2$ $2$ $5$
120.288.5-60.ke.2.4 $120$ $2$ $2$ $5$
120.288.5-60.lf.1.4 $120$ $2$ $2$ $5$
120.288.5-60.lf.2.8 $120$ $2$ $2$ $5$
120.288.5-120.dam.1.14 $120$ $2$ $2$ $5$
120.288.5-120.dam.2.14 $120$ $2$ $2$ $5$
120.288.5-120.dap.1.12 $120$ $2$ $2$ $5$
120.288.5-120.dap.2.12 $120$ $2$ $2$ $5$
120.288.5-120.dhk.1.6 $120$ $2$ $2$ $5$
120.288.5-120.dhk.2.6 $120$ $2$ $2$ $5$
120.288.5-120.dhn.1.16 $120$ $2$ $2$ $5$
120.288.5-120.dhn.2.16 $120$ $2$ $2$ $5$
120.288.9-30.a.1.10 $120$ $2$ $2$ $9$
120.288.9-60.b.1.4 $120$ $2$ $2$ $9$
120.288.9-30.n.1.2 $120$ $2$ $2$ $9$
120.288.9-30.q.1.7 $120$ $2$ $2$ $9$
120.288.9-30.q.2.8 $120$ $2$ $2$ $9$
120.288.9-30.t.1.8 $120$ $2$ $2$ $9$
120.288.9-30.t.2.8 $120$ $2$ $2$ $9$
120.288.9-30.u.1.4 $120$ $2$ $2$ $9$
120.288.9-30.v.1.8 $120$ $2$ $2$ $9$
120.288.9-60.db.1.6 $120$ $2$ $2$ $9$
120.288.9-60.ds.1.8 $120$ $2$ $2$ $9$
120.288.9-60.ds.2.6 $120$ $2$ $2$ $9$
120.288.9-60.ej.1.5 $120$ $2$ $2$ $9$
120.288.9-60.ej.2.6 $120$ $2$ $2$ $9$
120.288.9-60.em.1.8 $120$ $2$ $2$ $9$
120.288.9-60.ep.1.6 $120$ $2$ $2$ $9$
120.288.9-120.ipd.1.1 $120$ $2$ $2$ $9$
120.288.9-120.ipj.1.4 $120$ $2$ $2$ $9$
120.288.9-120.kvs.1.5 $120$ $2$ $2$ $9$
120.288.9-120.kvv.1.6 $120$ $2$ $2$ $9$
120.288.9-120.noi.1.8 $120$ $2$ $2$ $9$
120.288.9-120.noi.2.8 $120$ $2$ $2$ $9$
120.288.9-120.nol.1.12 $120$ $2$ $2$ $9$
120.288.9-120.nol.2.12 $120$ $2$ $2$ $9$
120.288.9-120.nxk.1.10 $120$ $2$ $2$ $9$
120.288.9-120.nxk.2.10 $120$ $2$ $2$ $9$
120.288.9-120.nxn.1.15 $120$ $2$ $2$ $9$
120.288.9-120.nxn.2.15 $120$ $2$ $2$ $9$
120.288.9-120.nxw.1.12 $120$ $2$ $2$ $9$
120.288.9-120.nxz.1.2 $120$ $2$ $2$ $9$
120.288.9-120.nyi.1.14 $120$ $2$ $2$ $9$
120.288.9-120.nyl.1.1 $120$ $2$ $2$ $9$
120.432.11-30.a.1.22 $120$ $3$ $3$ $11$