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: | 25.300.12.8 |
Level structure
| $\GL_2(\Z/25\Z)$-generators: | $\begin{bmatrix}4&15\\0&18\end{bmatrix}$, $\begin{bmatrix}16&8\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/25\Z)$-subgroup: | $C_{50}:C_{20}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 25.600.12-25.i.2.1, 25.600.12-25.i.2.2, 50.600.12-25.i.2.1, 50.600.12-25.i.2.2 |
| Cyclic 25-isogeny field degree: | $1$ |
| Cyclic 25-torsion field degree: | $10$ |
| 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^{2} + 2 x y + x z - 2 x w - x s - x b - x c - y^{2} - 2 y z - y t - y u + z w + z u + z b + s c $ |
| $=$ | $x z + x w - x u + x r - x c - y w + y t + y u - y v - y r + y s + y b + z^{2} + z t - 2 z v - z s + \cdots - s^{2}$ | |
| $=$ | $x z - 2 x t - 2 x u + x v + x r + x s - 2 x b - 2 x c - y z - y w - y r + y b - y c + 2 z u + 2 z b + \cdots + s c$ | |
| $=$ | $2 x^{2} + x y + x z - 2 x w + x t - 2 x r + x a + 2 x c - y^{2} - 2 y z - y w - y u - y v + y b + \cdots + b c$ | |
| $=$ | $\cdots$ |
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.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x-v-r$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x+a$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t+u+b+c$ |
| $\displaystyle W$ | $=$ | $\displaystyle x-y-z-2t-u+v+s-a-c$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}-XY-Y^{2}+3XZ+YZ+Z^{2}+XW-3YW-ZW-W^{2} $ |
| $=$ | $ X^{3}+2X^{2}Y-XY^{2}-Y^{3}+2X^{2}Z-3Y^{2}Z-XZ^{2}-2YZ^{2}-Z^{3}+X^{2}W-Y^{2}W+2XZW-2YZW-Z^{2}W+2XW^{2}+2ZW^{2}+W^{3} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 25.60.0.a.1 | $25$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 25.150.4.f.2 | $25$ | $2$ | $2$ | $4$ | $0$ | $8$ |
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.1 | $25$ | $5$ | $5$ | $76$ | $2$ | $2^{2}\cdot4^{5}\cdot8^{3}\cdot16$ |
| 50.600.37.i.2 | $50$ | $2$ | $2$ | $37$ | $0$ | $1^{3}\cdot2\cdot8\cdot12$ |
| 50.600.37.q.2 | $50$ | $2$ | $2$ | $37$ | $0$ | $1^{3}\cdot2\cdot8\cdot12$ |
| 50.900.48.i.2 | $50$ | $3$ | $3$ | $48$ | $0$ | $1^{2}\cdot2\cdot4^{2}\cdot8^{3}$ |
| 125.1500.76.c.2 | $125$ | $5$ | $5$ | $76$ | $?$ | not computed |