↑

Properties

Label 30.120.8.j.1
Level $30$
Index $120$
Genus $8$
Analytic rank $3$
Cusps $6$
$\Q$-cusps $2$

Related objects

Downloads

Learn more

Invariants

Level: $30$ $\SL_2$-level: $30$ Newform level: $900$
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 $10^{3}\cdot30^{3}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $3$
$\Q$-gonality: $3 \le \gamma \le 6$
$\overline{\Q}$-gonality: $3 \le \gamma \le 6$
Rational cusps: $2$
Rational CM points: yes $\quad(D =$ $-3$)

Other labels

Cummins and Pauli (CP) label: 30D8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 30.120.8.9

Level structure

$\GL_2(\Z/30\Z)$-generators: $\begin{bmatrix}5&11\\9&10\end{bmatrix}$, $\begin{bmatrix}19&10\\15&7\end{bmatrix}$, $\begin{bmatrix}20&17\\9&5\end{bmatrix}$, $\begin{bmatrix}23&15\\15&4\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 30.240.8-30.j.1.1, 30.240.8-30.j.1.2, 30.240.8-30.j.1.3, 30.240.8-30.j.1.4, 30.240.8-30.j.1.5, 30.240.8-30.j.1.6, 30.240.8-30.j.1.7, 30.240.8-30.j.1.8, 60.240.8-30.j.1.1, 60.240.8-30.j.1.2, 60.240.8-30.j.1.3, 60.240.8-30.j.1.4, 60.240.8-30.j.1.5, 60.240.8-30.j.1.6, 60.240.8-30.j.1.7, 60.240.8-30.j.1.8, 120.240.8-30.j.1.1, 120.240.8-30.j.1.2, 120.240.8-30.j.1.3, 120.240.8-30.j.1.4, 120.240.8-30.j.1.5, 120.240.8-30.j.1.6, 120.240.8-30.j.1.7, 120.240.8-30.j.1.8, 120.240.8-30.j.1.9, 120.240.8-30.j.1.10, 120.240.8-30.j.1.11, 120.240.8-30.j.1.12, 120.240.8-30.j.1.13, 120.240.8-30.j.1.14, 120.240.8-30.j.1.15, 120.240.8-30.j.1.16, 210.240.8-30.j.1.1, 210.240.8-30.j.1.2, 210.240.8-30.j.1.3, 210.240.8-30.j.1.4, 210.240.8-30.j.1.5, 210.240.8-30.j.1.6, 210.240.8-30.j.1.7, 210.240.8-30.j.1.8, 330.240.8-30.j.1.1, 330.240.8-30.j.1.2, 330.240.8-30.j.1.3, 330.240.8-30.j.1.4, 330.240.8-30.j.1.5, 330.240.8-30.j.1.6, 330.240.8-30.j.1.7, 330.240.8-30.j.1.8
Cyclic 30-isogeny field degree: $6$
Cyclic 30-torsion field degree: $48$
Full 30-torsion field degree: $1152$

Jacobian

Conductor: $2^{10}\cdot3^{13}\cdot5^{15}$
Simple: no
Squarefree: no
Decomposition: $1^{8}$
Newforms: 15.2.a.a, 75.2.a.a, 75.2.a.b, 900.2.a.a, 900.2.a.b$^{2}$, 900.2.a.e, 900.2.a.h

Models

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

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

Singular plane model Singular plane model

$ 0 $ $=$ $ 2560 x^{9} z^{2} + 14080 x^{8} y z^{2} + 21000 x^{7} y^{2} z^{2} - 3765 x^{7} z^{4} + \cdots + 3240 y^{7} z^{4} $
 Copy content Toggle raw display

Rational points

This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.

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

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 15.60.3.a.1 :

$\displaystyle X$ $=$ $\displaystyle x-2y-2z$
$\displaystyle Y$ $=$ $\displaystyle 4x-3y+2z$
$\displaystyle Z$ $=$ $\displaystyle x-2y+3z$

Equation of the image curve:

$0$ $=$ $ 2X^{4}+2X^{3}Y-9X^{2}Y^{2}+2XY^{3}+2Y^{4}+5X^{3}Z+2X^{2}YZ-2XY^{2}Z-5Y^{3}Z+4XYZ^{2}-7XZ^{3}+7YZ^{3}-4Z^{4} $

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

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
15.60.3.a.1 $15$ $2$ $2$ $3$ $0$ $1^{5}$
30.30.1.b.1 $30$ $4$ $4$ $1$ $1$ $1^{7}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
30.240.15.e.1 $30$ $2$ $2$ $15$ $3$ $1^{7}$
30.240.15.g.1 $30$ $2$ $2$ $15$ $6$ $1^{7}$
30.240.15.l.1 $30$ $2$ $2$ $15$ $4$ $1^{7}$
30.240.15.m.1 $30$ $2$ $2$ $15$ $5$ $1^{7}$
30.360.22.f.1 $30$ $3$ $3$ $22$ $4$ $1^{14}$
30.360.25.cq.1 $30$ $3$ $3$ $25$ $7$ $1^{17}$
60.240.15.cg.1 $60$ $2$ $2$ $15$ $6$ $1^{7}$
60.240.15.dc.1 $60$ $2$ $2$ $15$ $6$ $1^{7}$
60.240.15.et.1 $60$ $2$ $2$ $15$ $5$ $1^{7}$
60.240.15.ew.1 $60$ $2$ $2$ $15$ $5$ $1^{7}$
60.480.35.gz.1 $60$ $4$ $4$ $35$ $17$ $1^{27}$
120.240.15.xi.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.xr.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.zu.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.bad.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.bhm.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.bhv.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.bhy.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.bih.1 $120$ $2$ $2$ $15$ $?$ not computed
210.240.15.bl.1 $210$ $2$ $2$ $15$ $?$ not computed
210.240.15.bn.1 $210$ $2$ $2$ $15$ $?$ not computed
210.240.15.bp.1 $210$ $2$ $2$ $15$ $?$ not computed
210.240.15.br.1 $210$ $2$ $2$ $15$ $?$ not computed
330.240.15.bk.1 $330$ $2$ $2$ $15$ $?$ not computed
330.240.15.bm.1 $330$ $2$ $2$ $15$ $?$ not computed
330.240.15.bo.1 $330$ $2$ $2$ $15$ $?$ not computed
330.240.15.bq.1 $330$ $2$ $2$ $15$ $?$ not computed