↑

Properties

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

Related objects

Downloads

Learn more

Invariants

Level: $38$ $\SL_2$-level: $38$ Newform level: $76$
Index: $120$ $\PSL_2$-index:$120$
Genus: $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $2^{3}\cdot38^{3}$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4 \le \gamma \le 6$
$\overline{\Q}$-gonality: $4 \le \gamma \le 6$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 38A8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 38.120.8.1

Level structure

$\GL_2(\Z/38\Z)$-generators: $\begin{bmatrix}15&28\\0&31\end{bmatrix}$, $\begin{bmatrix}27&0\\0&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 38.240.8-38.a.1.1, 38.240.8-38.a.1.2, 76.240.8-38.a.1.1, 76.240.8-38.a.1.2, 76.240.8-38.a.1.3, 76.240.8-38.a.1.4, 76.240.8-38.a.1.5, 76.240.8-38.a.1.6, 76.240.8-38.a.1.7, 76.240.8-38.a.1.8, 76.240.8-38.a.1.9, 76.240.8-38.a.1.10, 114.240.8-38.a.1.1, 114.240.8-38.a.1.2, 152.240.8-38.a.1.1, 152.240.8-38.a.1.2, 152.240.8-38.a.1.3, 152.240.8-38.a.1.4, 152.240.8-38.a.1.5, 152.240.8-38.a.1.6, 152.240.8-38.a.1.7, 152.240.8-38.a.1.8, 152.240.8-38.a.1.9, 152.240.8-38.a.1.10, 152.240.8-38.a.1.11, 152.240.8-38.a.1.12, 190.240.8-38.a.1.1, 190.240.8-38.a.1.2, 228.240.8-38.a.1.1, 228.240.8-38.a.1.2, 228.240.8-38.a.1.3, 228.240.8-38.a.1.4, 228.240.8-38.a.1.5, 228.240.8-38.a.1.6, 228.240.8-38.a.1.7, 228.240.8-38.a.1.8, 228.240.8-38.a.1.9, 228.240.8-38.a.1.10, 266.240.8-38.a.1.1, 266.240.8-38.a.1.2
Cyclic 38-isogeny field degree: $1$
Cyclic 38-torsion field degree: $18$
Full 38-torsion field degree: $6156$

Jacobian

Conductor: $2^{6}\cdot19^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{8}$
Newforms: 19.2.a.a$^{3}$, 38.2.a.a$^{2}$, 38.2.a.b$^{2}$, 76.2.a.a

Models

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

$ 0 $ $=$ $ x^{2} + x t + x u + z t - t r + u^{2} + u v $
$=$ $x w - x u - z u - t^{2} + u v + u r$
$=$ $x v + y t + z t - t^{2} - t u - t v - t r + u v$
$=$ $x y + x w + x u - x r - y^{2} + y v + u^{2} + u r - v^{2} + v r$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 64 x^{12} - 352 x^{11} y + 960 x^{10} y^{2} - 80 x^{10} y z + 96 x^{10} z^{2} - 1724 x^{9} y^{3} + \cdots + y^{6} z^{6} $
 Copy content Toggle raw display

Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

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

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve $X_0(38)$ :

$\displaystyle X$ $=$ $\displaystyle x-y+r$
$\displaystyle Y$ $=$ $\displaystyle -x-t-u$
$\displaystyle Z$ $=$ $\displaystyle -x+y-z+r$
$\displaystyle W$ $=$ $\displaystyle -x+v-r$

Equation of the image curve:

$0$ $=$ $ X^{2}-XY+2XZ-YZ+2XW-YW+ZW $
$=$ $ Y^{3}-X^{2}Z+XZ^{2}+X^{2}W-XYW+YZW+2XW^{2}+YW^{2}-ZW^{2} $

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

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

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

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X(2)$ $2$ $20$ $20$ $0$ $0$ full Jacobian
$X_0(19)$ $19$ $6$ $6$ $1$ $0$ $1^{7}$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X(2)$ $2$ $20$ $20$ $0$ $0$ full Jacobian
38.40.2.a.1 $38$ $3$ $3$ $2$ $0$ $1^{6}$
$X_0(38)$ $38$ $2$ $2$ $4$ $0$ $1^{4}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
38.360.22.a.1 $38$ $3$ $3$ $22$ $0$ $2^{3}\cdot4^{2}$
38.360.22.a.2 $38$ $3$ $3$ $22$ $0$ $2^{3}\cdot4^{2}$
38.360.22.b.1 $38$ $3$ $3$ $22$ $4$ $1^{6}\cdot2^{4}$
$X_{\mathrm{sp}}(38)$ $38$ $19$ $19$ $161$ $56$ $1^{29}\cdot2^{23}\cdot3^{10}\cdot4^{7}\cdot6^{2}\cdot8$
76.240.16.a.1 $76$ $2$ $2$ $16$ $?$ not computed
76.240.16.a.2 $76$ $2$ $2$ $16$ $?$ not computed
76.240.17.a.1 $76$ $2$ $2$ $17$ $?$ not computed
76.240.17.b.1 $76$ $2$ $2$ $17$ $?$ not computed
76.240.17.c.1 $76$ $2$ $2$ $17$ $?$ not computed
76.240.17.d.1 $76$ $2$ $2$ $17$ $?$ not computed
76.240.18.a.1 $76$ $2$ $2$ $18$ $?$ not computed
76.240.18.a.2 $76$ $2$ $2$ $18$ $?$ not computed
152.240.16.a.1 $152$ $2$ $2$ $16$ $?$ not computed
152.240.16.a.2 $152$ $2$ $2$ $16$ $?$ not computed
152.240.17.a.1 $152$ $2$ $2$ $17$ $?$ not computed
152.240.17.b.1 $152$ $2$ $2$ $17$ $?$ not computed
152.240.17.c.1 $152$ $2$ $2$ $17$ $?$ not computed
152.240.17.d.1 $152$ $2$ $2$ $17$ $?$ not computed
152.240.18.a.1 $152$ $2$ $2$ $18$ $?$ not computed
152.240.18.a.2 $152$ $2$ $2$ $18$ $?$ not computed
228.240.16.a.1 $228$ $2$ $2$ $16$ $?$ not computed
228.240.16.a.2 $228$ $2$ $2$ $16$ $?$ not computed
228.240.17.a.1 $228$ $2$ $2$ $17$ $?$ not computed
228.240.17.b.1 $228$ $2$ $2$ $17$ $?$ not computed
228.240.17.c.1 $228$ $2$ $2$ $17$ $?$ not computed
228.240.17.d.1 $228$ $2$ $2$ $17$ $?$ not computed
228.240.18.a.1 $228$ $2$ $2$ $18$ $?$ not computed
228.240.18.a.2 $228$ $2$ $2$ $18$ $?$ not computed
266.360.22.a.1 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.a.2 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.b.1 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.b.2 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.c.1 $266$ $3$ $3$ $22$ $?$ not computed
266.360.22.c.2 $266$ $3$ $3$ $22$ $?$ not computed