Invariants
| Level: | $25$ | $\SL_2$-level: | $25$ | Newform level: | $125$ | ||
| Index: | $300$ | $\PSL_2$-index: | $300$ | ||||
| Genus: | $12 = 1 + \frac{ 300 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 28 }{2}$ | ||||||
| Cusps: | $28$ (none of which are rational) | Cusp widths | $1^{10}\cdot5^{8}\cdot25^{10}$ | Cusp orbits | $2^{5}\cdot4^{2}\cdot5^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 25B12 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 25.300.12.12 |
Level structure
| $\GL_2(\Z/25\Z)$-generators: | $\begin{bmatrix}1&17\\0&14\end{bmatrix}$, $\begin{bmatrix}18&19\\0&21\end{bmatrix}$ |
| $\GL_2(\Z/25\Z)$-subgroup: | $C_{50}:C_{20}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 25.600.12-25.i.1.1, 25.600.12-25.i.1.2, 50.600.12-25.i.1.1, 50.600.12-25.i.1.2 |
| Cyclic 25-isogeny field degree: | $1$ |
| Cyclic 25-torsion field degree: | $4$ |
| Full 25-torsion field degree: | $1000$ |
Jacobian
| Conductor: | $5^{32}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $4\cdot8$ |
| Newforms: | 25.2.d.a, 125.2.e.b |
Models
Canonical model in $\mathbb{P}^{ 11 }$ defined by 45 equations
| $ 0 $ | $=$ | $ x y + x t + x a + y v - t b - u a + v r - v b + a^{2} + a c $ |
| $=$ | $x y + x z + x u - x b + y z + y w - y s - w^{2} + w t + t^{2} + t v - t s - t c - r s + s b + s c$ | |
| $=$ | $x y - x w - x r - y^{2} - y z - y t + y v + w^{2} - t^{2} + t v + t b + v r - v b - v c - r a + r b - s a$ | |
| $=$ | $2 x z + x w + 2 x t + 2 x u + x v + x r - x s + x a - x b + y u - y v + z^{2} - w^{2} + t a - t b + \cdots - b c$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has no $\Q_p$ points for $p=2,3,13,17$, and therefore no rational points.
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 25.150.4.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z+w$ |
| $\displaystyle W$ | $=$ | $\displaystyle -y$ |
Equation of the image curve:
| $0$ | $=$ | $ Y^{2}-XZ+YZ-XW+ZW $ |
| $=$ | $ XYZ+X^{2}W-XYW-XZW+YZW+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 |
|---|---|---|---|---|---|---|
| 25.60.0.a.2 | $25$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 25.150.4.f.1 | $25$ | $2$ | $2$ | $4$ | $0$ | $8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 25.1500.76.c.2 | $25$ | $5$ | $5$ | $76$ | $2$ | $2^{2}\cdot4^{5}\cdot8^{3}\cdot16$ |
| 50.600.37.i.1 | $50$ | $2$ | $2$ | $37$ | $0$ | $1^{3}\cdot2\cdot8\cdot12$ |
| 50.600.37.q.1 | $50$ | $2$ | $2$ | $37$ | $0$ | $1^{3}\cdot2\cdot8\cdot12$ |
| 50.900.48.i.1 | $50$ | $3$ | $3$ | $48$ | $0$ | $1^{2}\cdot2\cdot4^{2}\cdot8^{3}$ |
| 125.1500.76.c.1 | $125$ | $5$ | $5$ | $76$ | $?$ | not computed |