Invariants
| Level: | $21$ | $\SL_2$-level: | $21$ | Newform level: | $441$ | ||
| Index: | $252$ | $\PSL_2$-index: | $252$ | ||||
| Genus: | $15 = 1 + \frac{ 252 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $21^{12}$ | Cusp orbits | $12$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 21B15 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 21.252.15.14 |
Level structure
| $\GL_2(\Z/21\Z)$-generators: | $\begin{bmatrix}9&5\\8&4\end{bmatrix}$, $\begin{bmatrix}17&17\\2&0\end{bmatrix}$ |
| $\GL_2(\Z/21\Z)$-subgroup: | $C_8\times C_{48}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 21-isogeny field degree: | $32$ |
| Cyclic 21-torsion field degree: | $384$ |
| Full 21-torsion field degree: | $384$ |
Jacobian
| Conductor: | $3^{28}\cdot7^{30}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{7}\cdot2^{4}$ |
| Newforms: | 49.2.a.a, 441.2.a.a, 441.2.a.b, 441.2.a.c, 441.2.a.d, 441.2.a.e, 441.2.a.f, 441.2.a.g, 441.2.a.h, 441.2.a.i, 441.2.a.j |
Models
Canonical model in $\mathbb{P}^{ 14 }$ defined by 78 equations
| $ 0 $ | $=$ | $ x^{2} - x a + y^{2} - y z + a^{2} $ |
| $=$ | $x^{2} + x z - x t - 3 x u + x v - x r + 3 x s - 2 x b - x c + 3 x d + x f + y^{2} + 2 y z + y w + \cdots - f^{2}$ | |
| $=$ | $x^{2} + x y + 4 x w + x t - 2 x u + 2 x v + 2 x r - x s + x a - 2 x b + 3 x d + x e + 3 x f - 2 y^{2} + \cdots - d f$ | |
| $=$ | $2 x^{2} + 4 x y - 3 x z - 4 x w + x u + 4 x v + x r - x s + x a - 2 x b + x d - 4 x e - 2 x f + y z + \cdots + e f$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has no real points, 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 $X_{\mathrm{ns}}^+(21)$ :
| $\displaystyle X$ | $=$ | $\displaystyle x-2a$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y+w+t-v+r-s+c-2d$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x+y+2w+2t-u-v+2r-s+c-2d-2f$ |
| $\displaystyle W$ | $=$ | $\displaystyle 2x+z+3w+t-u-2v+3r-s-2a+c-2d+2e-2f$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}+YZ-Z^{2}+XW+ZW-W^{2} $ |
| $=$ | $ X^{3}+2XY^{2}+Y^{3}+XYZ+Y^{2}Z-XZ^{2}-YZ^{2}-X^{2}W-2XYW-Y^{2}W+XZW+YZW+YW^{2} $ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(3)$ | $3$ | $42$ | $42$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}(7)$ | $7$ | $6$ | $6$ | $1$ | $0$ | $1^{6}\cdot2^{4}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(21)$ | $21$ | $2$ | $2$ | $4$ | $4$ | $1^{3}\cdot2^{4}$ |
| 21.126.6.a.1 | $21$ | $2$ | $2$ | $6$ | $1$ | $1^{5}\cdot2^{2}$ |
| 21.126.7.a.1 | $21$ | $2$ | $2$ | $7$ | $1$ | $1^{4}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(42)$ | $42$ | $2$ | $2$ | $37$ | $11$ | $1^{14}\cdot2^{2}\cdot4$ |
| 42.504.37.ba.1 | $42$ | $2$ | $2$ | $37$ | $12$ | $1^{14}\cdot2^{2}\cdot4$ |
| 42.504.37.ci.1 | $42$ | $2$ | $2$ | $37$ | $12$ | $1^{14}\cdot2^{2}\cdot4$ |
| 42.504.37.ck.1 | $42$ | $2$ | $2$ | $37$ | $11$ | $1^{14}\cdot2^{2}\cdot4$ |
| 42.756.51.a.1 | $42$ | $3$ | $3$ | $51$ | $15$ | $1^{20}\cdot2^{8}$ |
| 63.756.55.n.1 | $63$ | $3$ | $3$ | $55$ | $4$ | $2\cdot4^{2}\cdot6\cdot8^{3}$ |
| 63.756.55.p.1 | $63$ | $3$ | $3$ | $55$ | $4$ | $2\cdot4^{2}\cdot6\cdot8^{3}$ |
| 63.756.55.q.1 | $63$ | $3$ | $3$ | $55$ | $6$ | $2\cdot4^{2}\cdot6\cdot8^{3}$ |
| 63.756.55.bd.1 | $63$ | $3$ | $3$ | $55$ | $20$ | $1^{2}\cdot2^{4}\cdot3^{2}\cdot4^{6}$ |
| 63.756.55.bf.1 | $63$ | $3$ | $3$ | $55$ | $4$ | $2\cdot4^{2}\cdot6\cdot8^{3}$ |
| $X_{\mathrm{ns}}(63)$ | $63$ | $9$ | $9$ | $171$ | $74$ | $1^{6}\cdot2^{6}\cdot3^{6}\cdot4^{8}\cdot5^{4}\cdot6^{2}\cdot8^{2}\cdot12^{2}\cdot16$ |