Invariants
| Level: | $30$ | $\SL_2$-level: | $10$ | Newform level: | $180$ | ||
| 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: | 30.48.1.8 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}1&20\\5&9\end{bmatrix}$, $\begin{bmatrix}11&10\\12&23\end{bmatrix}$, $\begin{bmatrix}21&25\\23&14\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.24.1.d.2 for the level structure with $-I$) |
| Cyclic 30-isogeny field degree: | $12$ |
| Cyclic 30-torsion field degree: | $96$ |
| Full 30-torsion field degree: | $2880$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 180.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 372x + 2761 $ |
Rational points
This modular curve has rational points, including 2 rational_cusps and 1 known non-cuspidal non-CM point. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(11:0:1)$, $(0:1:0)$ | ||
| 75.a1 | no | $\tfrac{-102400}{3}$ | $= -1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{2}$ | $11.537$ | $(2:-45:1)$, $(2:45:1)$, $(12:-5:1)$, $(12:5:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{2184x^{2}y^{6}-49819344840x^{2}y^{4}z^{2}+3422526853924728x^{2}y^{2}z^{4}-27039497222900331576x^{2}z^{6}-1674672xy^{6}z+2876294021616xy^{4}z^{3}-100451964128318928xy^{2}z^{5}+602183388702393227664xz^{7}-y^{8}+510498348y^{6}z^{2}-106591444126974y^{4}z^{4}+1503562500387405180y^{2}z^{6}-3352238111755385915049z^{8}}{y^{2}(x^{2}y^{4}-594x^{2}y^{2}z^{2}-729x^{2}z^{4}+44xy^{4}z-6453xy^{2}z^{3}-8019xz^{5}+466y^{4}z^{2}+150876y^{2}z^{4}+182979z^{6})}$ |
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 |
|---|---|---|---|---|---|---|
| 5.24.0-5.a.2.2 | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 6.2.0.a.1 | $6$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 5.24.0-5.a.2.2 | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 30.24.0-5.a.2.1 | $30$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 30.144.1-30.d.2.4 | $30$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 30.144.5-30.k.1.2 | $30$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.144.5-30.l.1.2 | $30$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.192.5-30.c.2.1 | $30$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.240.5-30.e.1.4 | $30$ | $5$ | $5$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.192.5-60.v.1.7 | $60$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 150.240.5-150.b.2.3 | $150$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 210.384.13-210.f.2.2 | $210$ | $8$ | $8$ | $13$ | $?$ | not computed |