Invariants
| Level: | $30$ | $\SL_2$-level: | $30$ | Newform level: | $900$ | ||
| Index: | $216$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $9 = 1 + \frac{ 216 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $6^{6}\cdot30^{6}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30R9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.216.9.10 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}1&15\\6&13\end{bmatrix}$, $\begin{bmatrix}7&5\\8&17\end{bmatrix}$, $\begin{bmatrix}23&20\\10&17\end{bmatrix}$ |
| $\GL_2(\Z/30\Z)$-subgroup: | $D_{20}:C_4^2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 30-isogeny field degree: | $4$ |
| Cyclic 30-torsion field degree: | $32$ |
| Full 30-torsion field degree: | $640$ |
Jacobian
| Conductor: | $2^{7}\cdot3^{16}\cdot5^{12}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 30.2.c.a, 45.2.b.a, 90.2.a.a, 90.2.a.b, 225.2.a.b, 450.2.a.d, 900.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x s - w t $ |
| $=$ | $x y + 2 x z - y t + z t$ | |
| $=$ | $x y + x u - x r - x s - y z + y u + y v + z^{2} - z u - z v$ | |
| $=$ | $x y + x u + x r + x s - y^{2} - y w + y t - y u + y v - z^{2} + z u + z v + t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 81328 x^{16} - 93600 x^{15} y + 320536 x^{15} z + 46800 x^{14} y^{2} - 468900 x^{14} y z + \cdots + 144 z^{16} $ |
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 |
|---|
| $(0:1/3:-1/3:0:0:-1/3:0:1:0)$, $(0:-1/2:1/2:0:0:1/2:0:1:0)$ |
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle r$ |
| $\displaystyle Z$ | $=$ | $\displaystyle s$ |
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 30.108.4.a.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y-z$ |
| $\displaystyle W$ | $=$ | $\displaystyle z-u$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}+XY-Y^{2}+Z^{2}-ZW-W^{2} $ |
| $=$ | $ X^{3}+X^{2}Y-XY^{2}-3X^{2}Z+XYZ-2XZ^{2}-YZ^{2}+Z^{3}-X^{2}W+2XYW+XZW-2YZW-Z^{2}W-ZW^{2} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.72.1.o.1 | $30$ | $3$ | $3$ | $1$ | $1$ | $1^{4}\cdot2^{2}$ |
| 30.108.4.a.2 | $30$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 30.108.4.c.1 | $30$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 30.108.5.f.1 | $30$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.432.25.h.1 | $30$ | $2$ | $2$ | $25$ | $1$ | $1^{8}\cdot2^{4}$ |
| 30.432.25.r.2 | $30$ | $2$ | $2$ | $25$ | $1$ | $1^{8}\cdot2^{4}$ |
| 30.432.25.bb.2 | $30$ | $2$ | $2$ | $25$ | $2$ | $1^{8}\cdot2^{4}$ |
| 30.432.25.bi.1 | $30$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 30.1080.69.e.1 | $30$ | $5$ | $5$ | $69$ | $8$ | $1^{26}\cdot2^{17}$ |
| 60.432.21.bo.2 | $60$ | $2$ | $2$ | $21$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.432.21.bv.2 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 60.432.21.fh.2 | $60$ | $2$ | $2$ | $21$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.432.21.fn.2 | $60$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 60.432.21.he.2 | $60$ | $2$ | $2$ | $21$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.432.21.hj.2 | $60$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 60.432.21.il.2 | $60$ | $2$ | $2$ | $21$ | $2$ | $1^{6}\cdot2^{3}$ |
| 60.432.21.ip.2 | $60$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 60.432.25.ch.2 | $60$ | $2$ | $2$ | $25$ | $5$ | $1^{8}\cdot2^{4}$ |
| 60.432.25.in.2 | $60$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 60.432.25.mk.2 | $60$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 60.432.25.qe.2 | $60$ | $2$ | $2$ | $25$ | $2$ | $1^{8}\cdot2^{4}$ |
| 60.432.29.bff.2 | $60$ | $2$ | $2$ | $29$ | $4$ | $1^{10}\cdot2^{5}$ |
| 60.432.29.bfj.2 | $60$ | $2$ | $2$ | $29$ | $4$ | $1^{10}\cdot2^{5}$ |
| 60.432.29.byg.2 | $60$ | $2$ | $2$ | $29$ | $7$ | $1^{10}\cdot2^{5}$ |
| 60.432.29.byl.2 | $60$ | $2$ | $2$ | $29$ | $2$ | $1^{10}\cdot2^{5}$ |
| 60.432.29.cpd.2 | $60$ | $2$ | $2$ | $29$ | $4$ | $1^{10}\cdot2^{5}$ |
| 60.432.29.cpj.2 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{10}\cdot2^{5}$ |
| 60.432.29.czs.2 | $60$ | $2$ | $2$ | $29$ | $5$ | $1^{10}\cdot2^{5}$ |
| 60.432.29.czz.2 | $60$ | $2$ | $2$ | $29$ | $3$ | $1^{10}\cdot2^{5}$ |
| 120.432.21.fo.2 | $120$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 120.432.21.hl.2 | $120$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 120.432.21.bfs.1 | $120$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 120.432.21.bhi.1 | $120$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 120.432.21.bsy.2 | $120$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 120.432.21.buh.2 | $120$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 120.432.21.cbw.2 | $120$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 120.432.21.ccy.2 | $120$ | $2$ | $2$ | $21$ | $?$ | not computed |