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Properties

Label 38.120.8.c.1
Level $38$
Index $120$
Genus $8$
Analytic rank $2$
Cusps $6$
$\Q$-cusps $0$

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Invariants

Level: $38$ $\SL_2$-level: $38$ Newform level: $1444$
Index: $120$ $\PSL_2$-index:$120$
Genus: $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (none of which are rational) Cusp widths $2^{3}\cdot38^{3}$ Cusp orbits $3^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-gonality: $3 \le \gamma \le 6$
$\overline{\Q}$-gonality: $3 \le \gamma \le 6$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 38B8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 38.120.8.6

Level structure

$\GL_2(\Z/38\Z)$-generators: $\begin{bmatrix}10&15\\25&37\end{bmatrix}$, $\begin{bmatrix}34&5\\19&33\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 38.240.8-38.c.1.1, 38.240.8-38.c.1.2, 76.240.8-38.c.1.1, 76.240.8-38.c.1.2, 76.240.8-38.c.1.3, 76.240.8-38.c.1.4, 76.240.8-38.c.1.5, 76.240.8-38.c.1.6, 114.240.8-38.c.1.1, 114.240.8-38.c.1.2, 152.240.8-38.c.1.1, 152.240.8-38.c.1.2, 152.240.8-38.c.1.3, 152.240.8-38.c.1.4, 152.240.8-38.c.1.5, 152.240.8-38.c.1.6, 152.240.8-38.c.1.7, 152.240.8-38.c.1.8, 190.240.8-38.c.1.1, 190.240.8-38.c.1.2, 228.240.8-38.c.1.1, 228.240.8-38.c.1.2, 228.240.8-38.c.1.3, 228.240.8-38.c.1.4, 228.240.8-38.c.1.5, 228.240.8-38.c.1.6, 266.240.8-38.c.1.1, 266.240.8-38.c.1.2
Cyclic 38-isogeny field degree: $3$
Cyclic 38-torsion field degree: $54$
Full 38-torsion field degree: $6156$

Jacobian

Conductor: $2^{8}\cdot19^{14}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot2^{2}$
Newforms: 19.2.a.a, 76.2.a.a, 722.2.a.c$^{2}$, 722.2.a.j$^{2}$

Models

Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations

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

$ 0 $ $=$ $ - 684 x^{12} - 1140 x^{11} y + 228 x^{11} z - 380 x^{10} y^{2} - 1444 x^{10} y z + 160 x^{10} z^{2} + \cdots + 7 y^{3} z^{9} $
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Rational points

This modular curve has no $\Q_p$ points for $p=5,41$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

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

Maps to other modular curves

$j$-invariant map of degree 120 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 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Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
38.40.2.a.1 $38$ $3$ $3$ $2$ $0$ $1^{2}\cdot2^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
38.360.22.b.1 $38$ $3$ $3$ $22$ $4$ $1^{10}\cdot2^{2}$
38.360.22.c.1 $38$ $3$ $3$ $22$ $4$ $2^{5}\cdot4$
38.360.22.c.2 $38$ $3$ $3$ $22$ $4$ $2^{5}\cdot4$
38.2280.161.e.1 $38$ $19$ $19$ $161$ $58$ $1^{29}\cdot2^{23}\cdot3^{10}\cdot4^{7}\cdot6^{2}\cdot8$
76.240.18.e.1 $76$ $2$ $2$ $18$ $?$ not computed
76.240.18.e.2 $76$ $2$ $2$ $18$ $?$ not computed
76.240.18.f.1 $76$ $2$ $2$ $18$ $?$ not computed
76.240.18.f.2 $76$ $2$ $2$ $18$ $?$ not computed
152.240.18.i.1 $152$ $2$ $2$ $18$ $?$ not computed
152.240.18.i.2 $152$ $2$ $2$ $18$ $?$ not computed
152.240.18.j.1 $152$ $2$ $2$ $18$ $?$ not computed
152.240.18.j.2 $152$ $2$ $2$ $18$ $?$ not computed
228.240.18.e.1 $228$ $2$ $2$ $18$ $?$ not computed
228.240.18.e.2 $228$ $2$ $2$ $18$ $?$ not computed
228.240.18.f.1 $228$ $2$ $2$ $18$ $?$ not computed
228.240.18.f.2 $228$ $2$ $2$ $18$ $?$ not computed
266.360.22.e.1 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.e.2 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.l.1 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.l.2 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.m.1 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.m.2 $266$ $3$ $3$ $22$ $?$ not computed