LMFDB - Modular curve 38.120.8.e.1

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:	$6$ (of which $2$ are rational)	Cusp widths	$2^{3}\cdot38^{3}$	Cusp orbits	$1^{2}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$2$
$\Q$-gonality:	$4 \le \gamma \le 6$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 6$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	38A8
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	38.120.8.3

Level structure

$\GL_2(\Z/38\Z)$-generators:	$\begin{bmatrix}31&18\\0&15\end{bmatrix}$, $\begin{bmatrix}33&13\\0&35\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	38.240.8-38.e.1.1, 38.240.8-38.e.1.2, 76.240.8-38.e.1.1, 76.240.8-38.e.1.2, 76.240.8-38.e.1.3, 76.240.8-38.e.1.4, 76.240.8-38.e.1.5, 76.240.8-38.e.1.6, 114.240.8-38.e.1.1, 114.240.8-38.e.1.2, 152.240.8-38.e.1.1, 152.240.8-38.e.1.2, 152.240.8-38.e.1.3, 152.240.8-38.e.1.4, 152.240.8-38.e.1.5, 152.240.8-38.e.1.6, 152.240.8-38.e.1.7, 152.240.8-38.e.1.8, 190.240.8-38.e.1.1, 190.240.8-38.e.1.2, 228.240.8-38.e.1.1, 228.240.8-38.e.1.2, 228.240.8-38.e.1.3, 228.240.8-38.e.1.4, 228.240.8-38.e.1.5, 228.240.8-38.e.1.6, 266.240.8-38.e.1.1, 266.240.8-38.e.1.2
Cyclic 38-isogeny field degree:	$1$
Cyclic 38-torsion field degree:	$18$
Full 38-torsion field degree:	$6156$

Jacobian

Conductor:	$2^{6}\cdot19^{12}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{8}$
Newforms:	19.2.a.a$^{2}$, 38.2.a.a, 38.2.a.b, 361.2.a.b, 722.2.a.b, 722.2.a.e, 1444.2.a.a

Models

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

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

[x*v + x*r - z^2 + 2*z*r - w*t + w*u - w*v, x*w - x*t - x*u + z*t - z*u - z*r + w*t - w*u, x*t - x*u + x*v + y*z + y*t - y*u - y*r + z*t, x*z + x*u - x*v - x*r - y*w + 2*y*u, x*y + x*v + y*z - y*t - y*u + y*r + z^2 + z*u - z*r - w*t + w*v + w*r, x*y + x*t + y*z - 2*y*v + z*t + z*u - z*v - w*t + w*u - w*v, x^2 + x*z - x*t - x*v + x*r + z*w - z*t - z*u + z*r + w*t - w*v - w*r, 5*x^2 - 4*x*y + x*z + 4*x*w + 2*x*t + x*u - x*v - x*r + y*z + 4*y*w + y*u - y*v + z*u + z*r - w*t + w*u - t^2 + 2*t*r - v*r - r^2, 2*x^2 + x*y - 2*x*z + 4*x*w + x*t - 2*x*v + 2*y*z - y*w + z^2 - 3*z*w - z*t - z*u + z*v + z*r - 5*w^2 + w*t + w*u - w*r - t^2 - t*v + 2*t*r - u^2 + u*v - v*r - r^2, 2*x^2 - 2*x*y + x*z - 3*x*w - x*t - x*u + x*v - 4*y*z + y*w + 2*y*u + 6*z^2 + 2*z*w + 2*z*t - z*u + z*r - w*t - w*v + w*r + t*v + u*v - v*r, x^2 + 2*x*y + 6*x*z + 4*x*w + x*t + x*v - 6*y*z - 3*y*u - y*v - 2*z^2 + z*w + 2*z*u - z*v - 2*z*r + w*v - t*u + t*v - u*v + u*r, 8*x^2 + 3*x*y - 2*x*z - 4*x*w - x*u + x*v - x*r - 2*y*z - 4*y*w + y*t + y*v - y*r - z^2 - z*w + z*t - z*u - z*r + 5*w^2 + w*t - 2*w*u + u^2 + v*r, 2*x^2 - 2*x*y + 3*x*z - 4*x*w - 2*x*t + 2*x*r + 3*y*z + 3*y*w - 2*y*t + y*u - y*v + 2*y*r + z^2 - 8*z*w + z*t - 5*w^2 + w*u - w*v - t*u - t*v - t*r + u*v + r^2, 2*x^2 + 5*x*y + x*z - x*t + x*u - x*v - x*r + 3*y*z + 10*y*w + y*t + 4*y*u + 3*y*v - y*r + z^2 - 3*z*w - 2*z*u + 2*z*v + z*r + w*t - w*u + w*v + t^2 - t*v - 2*t*r + u*v + r^2, 2*x^2 - 7*x*y - 2*x*z - x*v + x*r - 19*y^2 + 3*y*z - y*w - y*t - 2*y*v + y*r + z^2 - 3*z*w + z*r - w*t + w*u - w*v + t^2 - 2*t*v - t*r + u*v]

Singular plane model Singular plane model

$ 0 $

$=$

$ - 9025 x^{13} - 53789 x^{12} y - 150233 x^{11} y^{2} - 10108 x^{11} y z - 262916 x^{10} y^{3} + \cdots + 16 y^{5} z^{8} $

[-9025*x^13 - 53789*x^12*y - 150233*x^11*y^2 - 10108*x^11*y*z - 262916*x^10*y^3 - 58995*x^10*y^2*z - 3249*x^10*y*z^2 - 339952*x^9*y^4 - 160047*x^9*y^3*z - 18962*x^9*y^2*z^2 - 343186*x^8*y^5 - 263953*x^8*y^4*z - 53056*x^8*y^3*z^2 - 722*x^8*y^2*z^3 - 263481*x^7*y^6 - 302647*x^7*y^5*z - 94505*x^7*y^4*z^2 - 3479*x^7*y^3*z^3 - 361*x^7*y^2*z^4 - 148351*x^6*y^7 - 249471*x^6*y^6*z - 128616*x^6*y^5*z^2 - 9126*x^6*y^4*z^3 - 246*x^6*y^3*z^4 - 61335*x^5*y^8 - 155889*x^5*y^7*z - 132817*x^5*y^6*z^2 - 12701*x^5*y^5*z^3 + 2333*x^5*y^4*z^4 - 8230*x^4*y^9 - 73604*x^4*y^8*z - 97959*x^4*y^7*z^2 - 16692*x^4*y^6*z^3 + 7592*x^4*y^5*z^4 + 312*x^4*y^4*z^5 - 1560*x^3*y^10 - 25724*x^3*y^9*z - 49785*x^3*y^8*z^2 - 13263*x^3*y^7*z^3 + 9439*x^3*y^6*z^4 + 652*x^3*y^5*z^5 - 8*x^3*y^4*z^6 + 102*x^2*y^11 - 3292*x^2*y^10*z - 17508*x^2*y^9*z^2 - 5910*x^2*y^8*z^3 + 5816*x^2*y^7*z^4 + 1452*x^2*y^6*z^5 - 372*x^2*y^5*z^6 + 204*x*y^11*z - 2108*x*y^10*z^2 - 1802*x*y^9*z^3 + 1630*x*y^8*z^4 + 972*x*y^7*z^5 - 364*x*y^6*z^6 + 102*y^11*z^2 - 376*y^10*z^3 + 252*y^9*z^4 + 232*y^8*z^5 - 76*y^7*z^6 - 48*y^6*z^7 + 16*y^5*z^8]

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle z$
$\displaystyle Z$	$=$	$\displaystyle t$

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 $X_0(38)$ :

$\displaystyle X$	$=$	$\displaystyle -z$
$\displaystyle Y$	$=$	$\displaystyle x$
$\displaystyle Z$	$=$	$\displaystyle w$
$\displaystyle W$	$=$	$\displaystyle x+y+z$

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

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
38.6.0.b.1	$38$	$20$	$20$	$0$	$0$	full Jacobian
38.40.2.b.1	$38$	$3$	$3$	$2$	$1$	$1^{6}$
$X_0(38)$	$38$	$2$	$2$	$4$	$0$	$1^{4}$

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.m.1	$38$	$3$	$3$	$22$	$2$	$2^{3}\cdot4^{2}$
38.360.22.m.2	$38$	$3$	$3$	$22$	$2$	$2^{3}\cdot4^{2}$
38.360.22.o.1	$38$	$3$	$3$	$22$	$8$	$1^{6}\cdot2^{4}$
38.2280.161.g.1	$38$	$19$	$19$	$161$	$58$	$1^{29}\cdot2^{23}\cdot3^{10}\cdot4^{7}\cdot6^{2}\cdot8$
76.240.17.i.1	$76$	$2$	$2$	$17$	$?$	not computed
76.240.17.k.1	$76$	$2$	$2$	$17$	$?$	not computed
76.240.17.m.1	$76$	$2$	$2$	$17$	$?$	not computed
76.240.17.o.1	$76$	$2$	$2$	$17$	$?$	not computed
152.240.17.z.1	$152$	$2$	$2$	$17$	$?$	not computed
152.240.17.bf.1	$152$	$2$	$2$	$17$	$?$	not computed
152.240.17.bs.1	$152$	$2$	$2$	$17$	$?$	not computed
152.240.17.by.1	$152$	$2$	$2$	$17$	$?$	not computed
228.240.17.u.1	$228$	$2$	$2$	$17$	$?$	not computed
228.240.17.w.1	$228$	$2$	$2$	$17$	$?$	not computed
228.240.17.bm.1	$228$	$2$	$2$	$17$	$?$	not computed
228.240.17.bo.1	$228$	$2$	$2$	$17$	$?$	not computed
266.360.22.cm.1	$266$	$3$	$3$	$22$	$?$	not computed
266.360.22.cm.2	$266$	$3$	$3$	$22$	$?$	not computed
266.360.22.cp.1	$266$	$3$	$3$	$22$	$?$	not computed
266.360.22.cp.2	$266$	$3$	$3$	$22$	$?$	not computed
266.360.22.cw.1	$266$	$3$	$3$	$22$	$?$	not computed
266.360.22.cw.2	$266$	$3$	$3$	$22$	$?$	not computed

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