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Properties

Label 30.144.9.bj.2
Level $30$
Index $144$
Genus $9$
Analytic rank $2$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $30$ $\SL_2$-level: $30$ Newform level: $900$
Index: $144$ $\PSL_2$-index:$144$
Genus: $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $6^{4}\cdot30^{4}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 30K9
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 30.144.9.47

Level structure

$\GL_2(\Z/30\Z)$-generators: $\begin{bmatrix}1&3\\0&7\end{bmatrix}$, $\begin{bmatrix}9&14\\25&27\end{bmatrix}$, $\begin{bmatrix}21&4\\20&9\end{bmatrix}$, $\begin{bmatrix}25&21\\27&26\end{bmatrix}$
$\GL_2(\Z/30\Z)$-subgroup: $C_2\times D_{60}:C_4$
Contains $-I$: yes
Quadratic refinements: 30.288.9-30.bj.2.1, 30.288.9-30.bj.2.2, 30.288.9-30.bj.2.3, 30.288.9-30.bj.2.4, 30.288.9-30.bj.2.5, 30.288.9-30.bj.2.6, 60.288.9-30.bj.2.1, 60.288.9-30.bj.2.2, 60.288.9-30.bj.2.3, 60.288.9-30.bj.2.4, 60.288.9-30.bj.2.5, 60.288.9-30.bj.2.6, 120.288.9-30.bj.2.1, 120.288.9-30.bj.2.2, 120.288.9-30.bj.2.3, 120.288.9-30.bj.2.4, 120.288.9-30.bj.2.5, 120.288.9-30.bj.2.6, 120.288.9-30.bj.2.7, 120.288.9-30.bj.2.8, 120.288.9-30.bj.2.9, 120.288.9-30.bj.2.10, 120.288.9-30.bj.2.11, 120.288.9-30.bj.2.12, 210.288.9-30.bj.2.1, 210.288.9-30.bj.2.2, 210.288.9-30.bj.2.3, 210.288.9-30.bj.2.4, 210.288.9-30.bj.2.5, 210.288.9-30.bj.2.6, 330.288.9-30.bj.2.1, 330.288.9-30.bj.2.2, 330.288.9-30.bj.2.3, 330.288.9-30.bj.2.4, 330.288.9-30.bj.2.5, 330.288.9-30.bj.2.6
Cyclic 30-isogeny field degree: $6$
Cyclic 30-torsion field degree: $48$
Full 30-torsion field degree: $960$

Jacobian

Conductor: $2^{12}\cdot3^{17}\cdot5^{13}$
Simple: no
Squarefree: no
Decomposition: $1^{5}\cdot2^{2}$
Newforms: 15.2.a.a, 45.2.b.a, 180.2.d.a, 900.2.a.b$^{2}$, 900.2.a.g$^{2}$

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

$ 0 $ $=$ $ w v + t r + u v $
$=$ $x v - y v + z r$
$=$ $x t + z t - z u$
$=$ $x v - y t + z u$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 328050 x^{16} - 271350 x^{14} y^{2} + 357075 x^{14} y z - 10576350 x^{14} z^{2} + \cdots + 590490 y^{2} z^{14} $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:-1/3:1/3:0:0:0:1)$, $(0:0:0:1/3:0:-1/3:0:0:1)$, $(0:0:0:1/2:-1/2:0:0:0:1)$, $(0:0:0:-1/2:0:1/2:0:0:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle s$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}r$

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 30.72.5.v.1 :

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

Equation of the image curve:

$0$ $=$ $ Y^{2}-ZW $
$=$ $ YZ+2YW+6W^{2}-YT+ZT-WT-T^{2} $
$=$ $ 15X^{2}-Y^{2}+YZ+5YW-ZW $

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
15.72.3.e.2 $15$ $2$ $2$ $3$ $0$ $1^{4}\cdot2$
30.72.1.o.1 $30$ $2$ $2$ $1$ $1$ $1^{4}\cdot2^{2}$
30.72.3.d.1 $30$ $2$ $2$ $3$ $0$ $1^{4}\cdot2$
30.72.5.v.1 $30$ $2$ $2$ $5$ $2$ $2^{2}$
30.72.5.ba.1 $30$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
30.72.5.bb.2 $30$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
30.72.5.bl.1 $30$ $2$ $2$ $5$ $1$ $1^{4}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
30.288.17.i.2 $30$ $2$ $2$ $17$ $3$ $1^{4}\cdot2^{2}$
30.288.17.s.2 $30$ $2$ $2$ $17$ $2$ $1^{4}\cdot2^{2}$
30.432.25.bi.1 $30$ $3$ $3$ $25$ $3$ $1^{8}\cdot2^{4}$
30.720.49.ff.1 $30$ $5$ $5$ $49$ $7$ $1^{18}\cdot2^{11}$
60.288.17.dv.1 $60$ $2$ $2$ $17$ $4$ $1^{4}\cdot2^{2}$
60.288.17.lo.2 $60$ $2$ $2$ $17$ $3$ $1^{4}\cdot2^{2}$
60.576.41.lp.1 $60$ $4$ $4$ $41$ $10$ $1^{16}\cdot2^{8}$
120.288.17.bzfv.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.bzgj.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.crhw.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.crik.2 $120$ $2$ $2$ $17$ $?$ not computed
210.288.17.ce.2 $210$ $2$ $2$ $17$ $?$ not computed
210.288.17.cu.1 $210$ $2$ $2$ $17$ $?$ not computed
330.288.17.cf.1 $330$ $2$ $2$ $17$ $?$ not computed
330.288.17.cv.1 $330$ $2$ $2$ $17$ $?$ not computed