Invariants
| Level: | $52$ | $\SL_2$-level: | $26$ | Newform level: | $2704$ | ||
| Index: | $182$ | $\PSL_2$-index: | $182$ | ||||
| Genus: | $10 = 1 + \frac{ 182 }{12} - \frac{ 0 }{4} - \frac{ 8 }{3} - \frac{ 7 }{2}$ | ||||||
| Cusps: | $7$ (none of which are rational) | Cusp widths | $26^{7}$ | Cusp orbits | $3\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $8$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 26A10 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 52.182.10.2 |
Level structure
| $\GL_2(\Z/52\Z)$-generators: | $\begin{bmatrix}36&5\\5&36\end{bmatrix}$, $\begin{bmatrix}47&28\\17&31\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 52-isogeny field degree: | $36$ |
| Cyclic 52-torsion field degree: | $864$ |
| Full 52-torsion field degree: | $13824$ |
Jacobian
| Conductor: | $2^{28}\cdot13^{19}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{4}\cdot3^{2}$ |
| Newforms: | 169.2.a.b, 208.2.a.c, 2704.2.a.a, 2704.2.a.g, 2704.2.a.m, 2704.2.a.x |
Models
Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations
| $ 0 $ | $=$ | $ x y - x w - y z + y u + y a - z t - w r - w s - t u - t s $ |
| $=$ | $z^{2} - z s - w^{2} - w t - t^{2} - v r - r s + r a$ | |
| $=$ | $x v + x s - x a + y w - y t - z^{2} + z r + w^{2}$ | |
| $=$ | $x u - x s - 2 y w - y t + z^{2} + z a + t^{2} - u r + v r + r s - r a$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6210750 x^{6} y^{12} + 36651875 x^{6} y^{11} z + 22593610 x^{6} y^{10} z^{2} - 127227763 x^{6} y^{9} z^{3} + \cdots + 4624 z^{18} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle a$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
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 $X_{S_4}(13)$ :
| $\displaystyle X$ | $=$ | $\displaystyle -y-w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w+t$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}-7X^{3}Y+3X^{2}Y^{2}+2XY^{3}+8X^{3}Z-7X^{2}YZ-2XY^{2}Z+5Y^{3}Z+4X^{2}Z^{2}-5XYZ^{2}+Y^{2}Z^{2}-3XZ^{3}+2YZ^{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 |
|---|---|---|---|---|---|---|
| $X_{S_4}(13)$ | $13$ | $2$ | $2$ | $3$ | $3$ | $1^{4}\cdot3$ |
| 52.2.0.a.1 | $52$ | $91$ | $91$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 52.546.36.k.1 | $52$ | $3$ | $3$ | $36$ | $20$ | $1^{14}\cdot3^{4}$ |
| 52.546.36.n.1 | $52$ | $3$ | $3$ | $36$ | $14$ | $1^{10}\cdot2^{2}\cdot3^{4}$ |
| 52.728.45.k.1 | $52$ | $4$ | $4$ | $45$ | $22$ | $1^{9}\cdot2^{4}\cdot3^{6}$ |
| 52.728.52.d.1 | $52$ | $4$ | $4$ | $52$ | $28$ | $1^{14}\cdot2^{5}\cdot3^{4}\cdot6$ |