Invariants
| Level: | $13$ | $\SL_2$-level: | $13$ | Newform level: | $169$ | ||
| Index: | $364$ | $\PSL_2$-index: | $364$ | ||||
| Genus: | $16 = 1 + \frac{ 364 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 28 }{2}$ | ||||||
| Cusps: | $28$ (none of which are rational) | Cusp widths | $13^{28}$ | Cusp orbits | $4\cdot12^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| Analytic rank: | $9$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 12$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 13A16 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 13.364.16.2 |
| Sutherland (S) label: | 13Ns.4.1 |
Level structure
| $\GL_2(\Z/13\Z)$-generators: | $\begin{bmatrix}0&12\\5&0\end{bmatrix}$, $\begin{bmatrix}0&12\\6&0\end{bmatrix}$ |
| $\GL_2(\Z/13\Z)$-subgroup: | $C_3:C_{24}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 13-isogeny field degree: | $2$ |
| Cyclic 13-torsion field degree: | $24$ |
| Full 13-torsion field degree: | $72$ |
Jacobian
| Conductor: | $13^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $2^{2}\cdot3^{4}$ |
| Newforms: | 169.2.a.a$^{2}$, 169.2.a.b$^{3}$, 169.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 15 }$ defined by 91 equations
| $ 0 $ | $=$ | $ 5 x y + 2 x z + 2 x t + x v + 3 x r - 2 x s - 3 x a + 2 x c + 3 x d - 2 x e + x f - 2 y^{2} + 7 y z + \cdots + g^{2} $ |
| $=$ | $x^{2} + x y - 3 x z + 4 x w + 5 x t + 6 x v - x r - x a + x b + 4 x c - 3 x d - 2 x f - 5 x g + \cdots + g^{2}$ | |
| $=$ | $3 x^{2} - 2 x y - 3 x z + 2 x w - x t - 2 x u - 3 x v + 5 x s - 3 x a + 3 x b - 4 x c - 3 x d + \cdots - g^{2}$ | |
| $=$ | $3 x^{2} - 7 x y + 6 x z + x w - x t + 2 x u + 5 x v - x r - 3 x s + x a - 5 x b + 3 x c - x d + \cdots + 2 g^{2}$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
Map of degree 4 from the canonical model of this modular curve to the canonical model of the modular curve $X_{S_4}(13)$ :
| $\displaystyle X$ | $=$ | $\displaystyle -x-z+w+a+b+d-f$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y-u-c$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x+y+z-w-2a-b-c-d-e+f$ |
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 |
|---|---|---|---|---|---|---|
| $X_{S_4}(13)$ | $13$ | $4$ | $4$ | $3$ | $3$ | $2^{2}\cdot3^{3}$ |
| 13.182.8.b.1 | $13$ | $2$ | $2$ | $8$ | $6$ | $2\cdot3^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 13.1092.50.d.1 | $13$ | $3$ | $3$ | $50$ | $21$ | $2^{5}\cdot3^{8}$ |
| 26.728.45.b.1 | $26$ | $2$ | $2$ | $45$ | $16$ | $1^{13}\cdot2^{2}\cdot3^{4}$ |
| 26.728.45.d.1 | $26$ | $2$ | $2$ | $45$ | $22$ | $1^{13}\cdot2^{2}\cdot3^{4}$ |
| 26.1092.64.b.1 | $26$ | $3$ | $3$ | $64$ | $26$ | $1^{20}\cdot2^{2}\cdot3^{8}$ |
| 39.1092.78.g.1 | $39$ | $3$ | $3$ | $78$ | $37$ | $1^{4}\cdot2^{7}\cdot4^{5}\cdot6^{4}$ |
| 39.1456.93.c.1 | $39$ | $4$ | $4$ | $93$ | $37$ | $1^{9}\cdot2^{16}\cdot3^{12}$ |
| 52.728.45.e.1 | $52$ | $2$ | $2$ | $45$ | $16$ | $1^{13}\cdot2^{2}\cdot3^{4}$ |
| 52.728.45.k.1 | $52$ | $2$ | $2$ | $45$ | $22$ | $1^{13}\cdot2^{2}\cdot3^{4}$ |
| 52.1456.107.s.1 | $52$ | $4$ | $4$ | $107$ | $59$ | $1^{9}\cdot2^{15}\cdot3^{4}\cdot4^{4}\cdot6^{4}$ |
| 65.1820.136.g.1 | $65$ | $5$ | $5$ | $136$ | $79$ | $1^{22}\cdot2^{5}\cdot5^{4}\cdot10^{2}\cdot12^{4}$ |
| 65.2184.155.k.1 | $65$ | $6$ | $6$ | $155$ | $63$ | $1^{5}\cdot2^{20}\cdot3^{14}\cdot4^{4}\cdot9^{4}$ |
| 65.3640.275.s.1 | $65$ | $10$ | $10$ | $275$ | $159$ | $1^{27}\cdot2^{25}\cdot3^{14}\cdot4^{4}\cdot5^{4}\cdot9^{4}\cdot10^{2}\cdot12^{4}$ |
| 169.4732.340.n.1 | $169$ | $13$ | $13$ | $340$ | $?$ | not computed |
| 169.61516.4944.b.1 | $169$ | $169$ | $169$ | $4944$ | $?$ | not computed |