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: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot30^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $4$ | ||||||
$\overline{\Q}$-gonality: | $4$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 30K9 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.144.9.61 |
Level structure
$\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}3&17\\29&0\end{bmatrix}$, $\begin{bmatrix}8&17\\7&11\end{bmatrix}$ |
$\GL_2(\Z/30\Z)$-subgroup: | $C_{60}.C_4^2$ |
Contains $-I$: | yes |
Quadratic refinements: | 60.288.9-30.h.1.1, 60.288.9-30.h.1.2, 60.288.9-30.h.1.3, 60.288.9-30.h.1.4, 120.288.9-30.h.1.1, 120.288.9-30.h.1.2, 120.288.9-30.h.1.3, 120.288.9-30.h.1.4 |
Cyclic 30-isogeny field degree: | $12$ |
Cyclic 30-torsion field degree: | $96$ |
Full 30-torsion field degree: | $960$ |
Jacobian
Conductor: | $2^{12}\cdot3^{14}\cdot5^{9}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{5}\cdot2^{2}$ |
Newforms: | 20.2.a.a, 36.2.a.a$^{2}$, 45.2.b.a, 60.2.d.a, 225.2.a.b, 900.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
$ 0 $ | $=$ | $ x w + z t $ |
$=$ | $x t - x r + t u$ | |
$=$ | $x w + z r + w u$ | |
$=$ | $y t + z w$ | |
$=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 81 x^{8} y^{4} z^{4} - 162 x^{8} y^{3} z^{5} - 891 x^{8} y^{2} z^{6} + 972 x^{8} y z^{7} + \cdots + 20124196 z^{16} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle s$ |
$\displaystyle Y$ | $=$ | $\displaystyle r$ |
$\displaystyle Z$ | $=$ | $\displaystyle t$ |
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 30.72.5.e.1 :
$\displaystyle X$ | $=$ | $\displaystyle -x$ |
$\displaystyle Y$ | $=$ | $\displaystyle -z$ |
$\displaystyle Z$ | $=$ | $\displaystyle -y$ |
$\displaystyle W$ | $=$ | $\displaystyle -w+t-2r$ |
$\displaystyle T$ | $=$ | $\displaystyle y-s$ |
Equation of the image curve:
$0$ | $=$ | $ Y^{2}-XZ $ |
$=$ | $ 75XY-15Y^{2}-15XZ+15YZ+W^{2} $ | |
$=$ | $ 94X^{2}+38XY-Y^{2}-2XZ+8YZ+4Z^{2}+XT+YT+ZT+T^{2} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
$X_{\mathrm{ns}}(2)$ | $2$ | $72$ | $72$ | $0$ | $0$ | full Jacobian |
15.72.3.h.2 | $15$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
15.72.3.h.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.f.1 | $30$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
30.72.5.b.1 | $30$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
30.72.5.e.1 | $30$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
30.72.5.g.2 | $30$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
30.72.5.bm.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 |
---|---|---|---|---|---|---|
30.432.25.h.1 | $30$ | $3$ | $3$ | $25$ | $1$ | $1^{8}\cdot2^{4}$ |
30.720.49.bg.1 | $30$ | $5$ | $5$ | $49$ | $6$ | $1^{18}\cdot2^{11}$ |
60.576.41.ea.1 | $60$ | $4$ | $4$ | $41$ | $8$ | $1^{16}\cdot2^{8}$ |