Invariants
Level: | $30$ | $\SL_2$-level: | $10$ | Newform level: | $20$ | ||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 10D1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.48.1.27 |
Level structure
$\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}16&15\\5&29\end{bmatrix}$, $\begin{bmatrix}22&25\\27&19\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 10.24.1.a.1 for the level structure with $-I$) |
Cyclic 30-isogeny field degree: | $12$ |
Cyclic 30-torsion field degree: | $48$ |
Full 30-torsion field degree: | $2880$ |
Jacobian
Conductor: | $2^{2}\cdot5$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 20.2.a.a |
Models
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} + x^{2} - 41x - 116 $ |
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.
Weierstrass model |
---|
$(-4:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{8x^{2}y^{6}+1560x^{2}y^{4}z^{2}+94264x^{2}y^{2}z^{4}+1875624x^{2}z^{6}+80xy^{6}z+13392xy^{4}z^{3}+777584xy^{2}z^{5}+15174128xz^{7}+y^{8}+332y^{6}z^{2}+37030y^{4}z^{4}+1771212y^{2}z^{6}+30686529z^{8}}{z^{3}y^{2}(122x^{2}z+xy^{2}+987xz^{2}+15y^{2}z+1996z^{3})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
$X_{\mathrm{ns}}(2)$ | $2$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
15.24.0-5.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
15.24.0-5.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
30.24.0-5.a.1.2 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
30.144.1-10.a.1.1 | $30$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
30.144.5-30.a.2.5 | $30$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
30.192.5-30.a.2.5 | $30$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
30.240.5-10.c.1.1 | $30$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
60.96.3-20.a.2.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
60.96.3-20.c.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
60.96.3-60.d.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
60.96.3-60.f.2.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
60.192.5-20.a.1.5 | $60$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
90.144.1-90.a.1.1 | $90$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
120.96.3-40.a.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.96.3-40.c.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.96.3-120.e.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.96.3-120.g.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
150.240.5-50.a.1.2 | $150$ | $5$ | $5$ | $5$ | $?$ | not computed |
210.144.1-70.b.2.1 | $210$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
210.384.13-70.a.2.2 | $210$ | $8$ | $8$ | $13$ | $?$ | not computed |