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Properties

Label 38.180.10.a.1
Level $38$
Index $180$
Genus $10$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $6$

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Invariants

Level: $38$ $\SL_2$-level: $38$ Newform level: $38$
Index: $180$ $\PSL_2$-index:$180$
Genus: $10 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $6$ are rational) Cusp widths $1^{3}\cdot2^{3}\cdot19^{3}\cdot38^{3}$ Cusp orbits $1^{6}\cdot3^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3 \le \gamma \le 6$
$\overline{\Q}$-gonality: $3 \le \gamma \le 6$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 38A10
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 38.180.10.1

Level structure

$\GL_2(\Z/38\Z)$-generators: $\begin{bmatrix}11&13\\0&13\end{bmatrix}$, $\begin{bmatrix}27&35\\0&5\end{bmatrix}$, $\begin{bmatrix}37&2\\0&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 38.360.10-38.a.1.1, 38.360.10-38.a.1.2, 38.360.10-38.a.1.3, 38.360.10-38.a.1.4, 76.360.10-38.a.1.1, 76.360.10-38.a.1.2, 76.360.10-38.a.1.3, 76.360.10-38.a.1.4, 76.360.10-38.a.1.5, 76.360.10-38.a.1.6, 76.360.10-38.a.1.7, 76.360.10-38.a.1.8, 76.360.10-38.a.1.9, 76.360.10-38.a.1.10, 76.360.10-38.a.1.11, 76.360.10-38.a.1.12, 114.360.10-38.a.1.1, 114.360.10-38.a.1.2, 114.360.10-38.a.1.3, 114.360.10-38.a.1.4, 152.360.10-38.a.1.1, 152.360.10-38.a.1.2, 152.360.10-38.a.1.3, 152.360.10-38.a.1.4, 152.360.10-38.a.1.5, 152.360.10-38.a.1.6, 152.360.10-38.a.1.7, 152.360.10-38.a.1.8, 152.360.10-38.a.1.9, 152.360.10-38.a.1.10, 152.360.10-38.a.1.11, 152.360.10-38.a.1.12, 152.360.10-38.a.1.13, 152.360.10-38.a.1.14, 152.360.10-38.a.1.15, 152.360.10-38.a.1.16, 190.360.10-38.a.1.1, 190.360.10-38.a.1.2, 190.360.10-38.a.1.3, 190.360.10-38.a.1.4, 228.360.10-38.a.1.1, 228.360.10-38.a.1.2, 228.360.10-38.a.1.3, 228.360.10-38.a.1.4, 228.360.10-38.a.1.5, 228.360.10-38.a.1.6, 228.360.10-38.a.1.7, 228.360.10-38.a.1.8, 228.360.10-38.a.1.9, 228.360.10-38.a.1.10, 228.360.10-38.a.1.11, 228.360.10-38.a.1.12, 266.360.10-38.a.1.1, 266.360.10-38.a.1.2, 266.360.10-38.a.1.3, 266.360.10-38.a.1.4
Cyclic 38-isogeny field degree: $1$
Cyclic 38-torsion field degree: $6$
Full 38-torsion field degree: $4104$

Jacobian

Conductor: $2^{8}\cdot19^{10}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot2\cdot4$
Newforms: 19.2.a.a$^{2}$, 38.2.a.a, 38.2.a.b, 38.2.c.a, 38.2.c.b

Models

Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations

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

$ 0 $ $=$ $ x^{6} y^{4} + 4 x^{5} y^{4} z + 2 x^{5} y^{2} z^{3} - 4 x^{5} y z^{4} + x^{5} z^{5} + 2 x^{4} y^{5} z + \cdots + y^{4} z^{6} $
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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
$(0:0:0:-1:1:1:0:0:0:0)$, $(0:0:0:0:0:1:0:0:0:0)$, $(0:0:1:0:1:0:0:0:0:0)$, $(0:0:0:0:0:0:0:-1:1:0)$, $(0:0:0:0:0:0:0:1:1:0)$, $(-2:0:0:0:0:0:0:1:1:0)$

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 3 from the canonical model of this modular curve to the canonical model of the modular curve $X_0(38)$ :

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

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

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Cover information

Click on a modular curve in the diagram to see information about it.
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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
$X_0(2)$ $2$ $60$ $60$ $0$ $0$ full Jacobian
19.60.1.a.2 $19$ $3$ $3$ $1$ $0$ $1^{3}\cdot2\cdot4$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
19.60.1.a.2 $19$ $3$ $3$ $1$ $0$ $1^{3}\cdot2\cdot4$
$X_0(38)$ $38$ $3$ $3$ $4$ $0$ $2\cdot4$

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 $38$ $2$ $2$ $22$ $0$ $1^{4}\cdot2^{2}\cdot4$
38.360.22.m.2 $38$ $2$ $2$ $22$ $2$ $1^{4}\cdot2^{2}\cdot4$
38.540.28.a.1 $38$ $3$ $3$ $28$ $0$ $6^{3}$
38.540.28.b.1 $38$ $3$ $3$ $28$ $0$ $6^{3}$
$X_{\pm1}(38)$ $38$ $3$ $3$ $28$ $0$ $6^{3}$
38.3420.226.a.1 $38$ $19$ $19$ $226$ $30$ $1^{14}\cdot2^{26}\cdot3^{6}\cdot4^{16}\cdot6^{6}\cdot8^{4}$
76.360.22.a.2 $76$ $2$ $2$ $22$ $?$ not computed
76.360.22.c.1 $76$ $2$ $2$ $22$ $?$ not computed
76.360.22.d.1 $76$ $2$ $2$ $22$ $?$ not computed
76.360.22.o.2 $76$ $2$ $2$ $22$ $?$ not computed
76.360.22.p.1 $76$ $2$ $2$ $22$ $?$ not computed
76.360.22.q.1 $76$ $2$ $2$ $22$ $?$ not computed
114.360.22.f.2 $114$ $2$ $2$ $22$ $?$ not computed
114.360.22.m.2 $114$ $2$ $2$ $22$ $?$ not computed
152.360.22.a.2 $152$ $2$ $2$ $22$ $?$ not computed
152.360.22.c.2 $152$ $2$ $2$ $22$ $?$ not computed
152.360.22.e.1 $152$ $2$ $2$ $22$ $?$ not computed
152.360.22.f.1 $152$ $2$ $2$ $22$ $?$ not computed
152.360.22.y.2 $152$ $2$ $2$ $22$ $?$ not computed
152.360.22.z.2 $152$ $2$ $2$ $22$ $?$ not computed
152.360.22.ba.1 $152$ $2$ $2$ $22$ $?$ not computed
152.360.22.bb.1 $152$ $2$ $2$ $22$ $?$ not computed
190.360.22.f.2 $190$ $2$ $2$ $22$ $?$ not computed
190.360.22.k.2 $190$ $2$ $2$ $22$ $?$ not computed
228.360.22.f.2 $228$ $2$ $2$ $22$ $?$ not computed
228.360.22.g.1 $228$ $2$ $2$ $22$ $?$ not computed
228.360.22.h.1 $228$ $2$ $2$ $22$ $?$ not computed
228.360.22.o.2 $228$ $2$ $2$ $22$ $?$ not computed
228.360.22.p.1 $228$ $2$ $2$ $22$ $?$ not computed
228.360.22.q.1 $228$ $2$ $2$ $22$ $?$ not computed
266.360.22.bm.2 $266$ $2$ $2$ $22$ $?$ not computed
266.360.22.co.1 $266$ $2$ $2$ $22$ $?$ not computed