LMFDB - Modular curve 195.336.11-65.a.1.8

Invariants

Level:	$195$	$\SL_2$-level:	$65$	Newform level:	$65$
Index:	$336$	$\PSL_2$-index:	$168$
Genus:	$11 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $4$ are rational)	Cusp widths	$1^{2}\cdot5^{2}\cdot13^{2}\cdot65^{2}$	Cusp orbits	$1^{4}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 11$
Rational cusps:	$4$
Rational CM points:	none

Level structure

$\GL_2(\Z/195\Z)$-generators:	$\begin{bmatrix}32&192\\160&104\end{bmatrix}$, $\begin{bmatrix}116&78\\9&125\end{bmatrix}$, $\begin{bmatrix}135&26\\158&163\end{bmatrix}$, $\begin{bmatrix}164&107\\35&1\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 65.168.11.a.1 for the level structure with $-I$)
Cyclic 195-isogeny field degree:	$4$
Cyclic 195-torsion field degree:	$192$
Full 195-torsion field degree:	$1797120$

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

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

[x*y - x*z + x*t + x*a - y*r + z*r - r*s - s^2 + s*a, x^2 - x*y + x*z + x*v + y*s - t^2 - t*a + s^2 - s*a, x^2 + x*r + x*s - x*a - y^2 + y*z - y*u - v*b - r*s - r*b - s^2 + s*a, 2*x^2 - x*y + x*z + x*t + x*s + x*b - y^2 + y*w - y*b + w^2 + t*b, x*z + x*t + x*u + x*r + x*b + z*v + t*b + u*b + r*b + b^2, x^2 + x*v - 2*x*r + x*a - y*r + y*a + u^2 - u*v, x*t + x*u + x*a + y*t + y*a + t*s + u*s + a*b, x*y - x*z + x*v + x*r - x*a - y*w - y*u + y*r - y*s - w^2 + t*s - t*b - s^2 + s*a, x*y + x*t + y^2 - y*z + y*t + y*b - t*r + t*s, y*u + y*b - t*u + t*v - t*r - t*s + t*a, x^2 + x*y - x*z + x*t + x*u + 2*x*b - t*u + t*v, x*y - x*a + y^2 - y*u + y*s - y*a - w*a + t*v + t*s - t*a, x*y - y*s - w^2 + t*u - t*v + t*s - t*b + u*s - v*s - r*b - s^2 + s*a + s*b, 2*x^2 + x*u + x*r - x*a - y*w + y*t + y*r + y*b - z*v - t^2 + t*s - t*a - u^2 + u*v - u*r + v*s, x^2 + x*z - x*s - y*s - w*a + t*u + t*b + u*b - s^2 + s*a - s*b + a*b + b^2, x*y + x*z - x*v + x*s - x*a + x*b - y*w + y*u - y*s - y*b + t*b - u^2 - u*r + u*s + v*b + r*s + r*b + s*b, x*y - x*u - x*v - x*s + y^2 - y*w - y*a - t^2 - t*u + t*v - t*a - u^2 - u*s + v*s + r*b - s*b, x^2 + x*z - x*u - 2*x*r + x*s + x*a + y*w + y*u - y*r - w*r + w*a + s*b - a*b, x*t - x*r + x*s + x*a + y*u - y*b + t*v + u*b + v*s + r*b, x^2 - x*z + x*w + x*u - 2*x*s + x*a - w*s - t^2 + t*r - t*a - t*b, x^2 + x*w + x*t - x*s + x*a + y*w + y*t + y*u + y*b + w*b + t*b, x*z - x*u - x*v + y*u - y*a + w^2 - w*s - w*b + t*u - t*s + t*b - u^2 + u*v + u*b + b^2, x*w - x*r + y*r - y*s - y*b - z*r + w*v + t*s - t*b + r*s, x*y + x*t + x*u - x*v + x*s + x*b - y*s - y*b + w*u + t*s - t*b - s^2 + s*a, x^2 + x*t - x*u - x*r + x*s + x*a + y*w + y*u - y*a - w*t + t*s - s^2 + s*a + s*b - a*b, 2*x^2 + x*z + x*v + x*s - x*a + x*b - y^2 + y*z + y*u + z*b - t*s + t*b - u*r + s*b - a*b, x*t + x*a + y*z + y*s - y*a + z*a - t*u + t*v, 2*x^2 + x*z - x*t - x*u - x*r - x*a + x*b - z*r - z*s - t^2 - t*a - u*v - v*b - r*b, x^2 - x*s + x*b + y*w + y*b - z*u + z*v - z*r + u^2 - u*v + u*r - v*s - v*b - r*b, x*t + x*a + y*w + y*t + y*u - z*t, x*w + x*t + x*r + x*s - y^2 + y*z - y*r + y*s - y*b + z*w + z*r - t*s + u*s - u*b - v*s - r*s - r*b + s*b - b^2, x*z - x*v - x*r - x*s + x*a + y^2 - y*z - y*r + z^2, x*y - x*z - x*u - x*r + x*b + y*v - y*s - y*b - z*v + t^2 + t*u + t*r + t*b + v*b + r*s, x*y - x*u + y*w + y*u - y*a + z*a - w*a - t*s + u*r - v*a, x*z + x*u - x*v + x*s + y*w + y*u - w*r + t^2 + 2*t*a + u*s + u*a - v*a + r*a + s*b + a*b, x*y + x*w + x*t + y*t - y*u - y*r + y*a + z*v + w*b - t*v + t*b + u*r - v^2 - v*r - v*b + r^2 - r*s - s*b + a*b]

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points.

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(13)$	$13$	$24$	$12$	$0$	$0$
15.24.0-5.a.1.2	$15$	$14$	$14$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
15.24.0-5.a.1.2	$15$	$14$	$14$	$0$	$0$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi
Siegel

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Invariants

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Level structure

Models

Rational points

Modular covers

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	195.336.11-65.a.1.8
Level	$195$
Index	$336$
Genus	$11$

Cusps	$8$
$\Q$-cusps	$4$