Invariants
| Level: | $28$ | $\SL_2$-level: | $28$ | Newform level: | $196$ | ||
| Index: | $336$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $9 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $3$ are rational) | Cusp widths | $14^{12}$ | Cusp orbits | $1^{3}\cdot3^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 14B9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 28.336.9.9 |
Level structure
| $\GL_2(\Z/28\Z)$-generators: | $\begin{bmatrix}3&0\\6&25\end{bmatrix}$, $\begin{bmatrix}3&10\\10&11\end{bmatrix}$, $\begin{bmatrix}7&16\\16&25\end{bmatrix}$, $\begin{bmatrix}7&18\\18&1\end{bmatrix}$, $\begin{bmatrix}25&10\\12&3\end{bmatrix}$ |
| $\GL_2(\Z/28\Z)$-subgroup: | $C_6^2:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 14.168.9.a.1 for the level structure with $-I$) |
| Cyclic 28-isogeny field degree: | $4$ |
| Cyclic 28-torsion field degree: | $48$ |
| Full 28-torsion field degree: | $576$ |
Jacobian
| Conductor: | $2^{12}\cdot7^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{3}$ |
| Newforms: | 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x y - x w + x v - x r - x s + y s - z s + u s - v s + r s $ |
| $=$ | $x y - x w + x v + x r + x s - y s + z s + t r - v s + r s$ | |
| $=$ | $y^{2} - y z - y w + y v + z w - z v + t r + u s - v r - v s + 2 r s$ | |
| $=$ | $2 x w - y w + z w + t v - u v - 2 v r + 2 v s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{13} y^{3} - 16 x^{13} y^{2} z - 8 x^{13} y z^{2} + 8 x^{13} z^{3} - 128 x^{12} y^{4} + \cdots + 8 y^{3} z^{13} $ |
Rational points
This modular curve has 3 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1:-1:-1:0:2:0:2:1:0)$, $(1:1:-1:2:0:0:1:1:1)$, $(1:-1:-1:0:0:2:2:0:1)$ |
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 14.84.3.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -4x+2y-2z-t+u+5r-5s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x-y+z+4t-4u-6r+6s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -4x+2y-2z-t+u-2r+2s$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}Y^{2}+X^{3}Z+4XY^{2}Z+Y^{3}Z+2X^{2}Z^{2}-XYZ^{2}-2XZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 14.168.9.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle r$ |
| $\displaystyle Y$ | $=$ | $\displaystyle v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle s$ |
Equation of the image curve:
| $0$ | $=$ | $ 8X^{13}Y^{3}-128X^{12}Y^{4}+86X^{11}Y^{5}+6X^{10}Y^{6}-9X^{9}Y^{7}-16X^{13}Y^{2}Z+326X^{12}Y^{3}Z-634X^{11}Y^{4}Z+511X^{10}Y^{5}Z+25X^{9}Y^{6}Z-75X^{8}Y^{7}Z-8X^{13}YZ^{2}+12X^{12}Y^{2}Z^{2}+510X^{11}Y^{3}Z^{2}-1537X^{10}Y^{4}Z^{2}+625X^{9}Y^{5}Z^{2}+148X^{8}Y^{6}Z^{2}-190X^{7}Y^{7}Z^{2}+8X^{13}Z^{3}-174X^{12}YZ^{3}+118X^{11}Y^{2}Z^{3}+879X^{10}Y^{3}Z^{3}+492X^{9}Y^{4}Z^{3}-1001X^{8}Y^{5}Z^{3}+488X^{7}Y^{6}Z^{3}-70X^{6}Y^{7}Z^{3}+118X^{12}Z^{4}-1196X^{11}YZ^{4}+3508X^{10}Y^{2}Z^{4}-7949X^{9}Y^{3}Z^{4}+7488X^{8}Y^{4}Z^{4}-2027X^{7}Y^{5}Z^{4}-130X^{6}Y^{6}Z^{4}+296X^{5}Y^{7}Z^{4}+564X^{11}Z^{5}-3843X^{10}YZ^{5}+10440X^{9}Y^{2}Z^{5}-10549X^{8}Y^{3}Z^{5}+318X^{7}Y^{4}Z^{5}+2046X^{6}Y^{5}Z^{5}-978X^{5}Y^{6}Z^{5}+296X^{4}Y^{7}Z^{5}+832X^{10}Z^{6}-1771X^{9}YZ^{6}-3224X^{8}Y^{2}Z^{6}+15863X^{7}Y^{3}Z^{6}-12094X^{6}Y^{4}Z^{6}+2046X^{5}Y^{5}Z^{6}-130X^{4}Y^{6}Z^{6}-70X^{3}Y^{7}Z^{6}-652X^{9}Z^{7}+6944X^{8}YZ^{7}-19852X^{7}Y^{2}Z^{7}+15863X^{6}Y^{3}Z^{7}+318X^{5}Y^{4}Z^{7}-2027X^{4}Y^{5}Z^{7}+488X^{3}Y^{6}Z^{7}-190X^{2}Y^{7}Z^{7}-2028X^{8}Z^{8}+6944X^{7}YZ^{8}-3224X^{6}Y^{2}Z^{8}-10549X^{5}Y^{3}Z^{8}+7488X^{4}Y^{4}Z^{8}-1001X^{3}Y^{5}Z^{8}+148X^{2}Y^{6}Z^{8}-75XY^{7}Z^{8}-652X^{7}Z^{9}-1771X^{6}YZ^{9}+10440X^{5}Y^{2}Z^{9}-7949X^{4}Y^{3}Z^{9}+492X^{3}Y^{4}Z^{9}+625X^{2}Y^{5}Z^{9}+25XY^{6}Z^{9}-9Y^{7}Z^{9}+832X^{6}Z^{10}-3843X^{5}YZ^{10}+3508X^{4}Y^{2}Z^{10}+879X^{3}Y^{3}Z^{10}-1537X^{2}Y^{4}Z^{10}+511XY^{5}Z^{10}+6Y^{6}Z^{10}+564X^{5}Z^{11}-1196X^{4}YZ^{11}+118X^{3}Y^{2}Z^{11}+510X^{2}Y^{3}Z^{11}-634XY^{4}Z^{11}+86Y^{5}Z^{11}+118X^{4}Z^{12}-174X^{3}YZ^{12}+12X^{2}Y^{2}Z^{12}+326XY^{3}Z^{12}-128Y^{4}Z^{12}+8X^{3}Z^{13}-8X^{2}YZ^{13}-16XY^{2}Z^{13}+8Y^{3}Z^{13} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 28.12.0-2.a.1.1 | $28$ | $28$ | $28$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 28.672.17-14.a.1.4 | $28$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 28.672.17-28.a.1.1 | $28$ | $2$ | $2$ | $17$ | $6$ | $1^{8}$ |
| 28.672.17-14.b.1.3 | $28$ | $2$ | $2$ | $17$ | $0$ | $1^{8}$ |
| 28.672.17-28.b.1.1 | $28$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 28.672.21-28.a.1.1 | $28$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.a.1.8 | $28$ | $2$ | $2$ | $21$ | $5$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.b.1.3 | $28$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.b.1.6 | $28$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.b.1.9 | $28$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.c.1.1 | $28$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.c.1.3 | $28$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.c.1.8 | $28$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.d.1.1 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.d.1.4 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.d.1.7 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 28.672.21-28.e.1.1 | $28$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 28.672.21-28.e.1.3 | $28$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 28.672.21-28.e.1.6 | $28$ | $2$ | $2$ | $21$ | $2$ | $1^{10}\cdot2$ |
| 28.672.21-28.f.1.2 | $28$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 28.672.21-28.f.1.3 | $28$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 28.672.21-28.f.1.6 | $28$ | $2$ | $2$ | $21$ | $5$ | $1^{10}\cdot2$ |
| 28.672.21-28.g.1.1 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 28.672.21-28.g.1.4 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 28.672.21-28.g.1.6 | $28$ | $2$ | $2$ | $21$ | $3$ | $1^{10}\cdot2$ |
| 28.672.21-28.h.1.2 | $28$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 28.672.21-28.h.1.4 | $28$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 28.672.21-28.h.1.6 | $28$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 28.1008.25-14.a.1.3 | $28$ | $3$ | $3$ | $25$ | $1$ | $1^{10}\cdot2^{3}$ |
| 56.672.17-56.a.1.4 | $56$ | $2$ | $2$ | $17$ | $6$ | $1^{8}$ |
| 56.672.17-56.b.1.4 | $56$ | $2$ | $2$ | $17$ | $0$ | $1^{8}$ |
| 56.672.17-56.c.1.4 | $56$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 56.672.17-56.d.1.4 | $56$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 56.672.21-56.a.1.5 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.a.1.12 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.a.1.16 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.b.1.5 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.b.1.12 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.b.1.16 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.c.1.8 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.c.1.10 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.c.1.15 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.d.1.8 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.d.1.10 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.d.1.15 | $56$ | $2$ | $2$ | $21$ | $6$ | $1^{6}\cdot2^{3}$ |
| 56.672.21-56.e.1.7 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.672.21-56.e.1.9 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.672.21-56.e.1.16 | $56$ | $2$ | $2$ | $21$ | $7$ | $1^{10}\cdot2$ |
| 56.672.21-56.f.1.7 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.f.1.9 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.f.1.16 | $56$ | $2$ | $2$ | $21$ | $4$ | $1^{10}\cdot2$ |
| 56.672.21-56.g.1.6 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{10}\cdot2$ |
| 56.672.21-56.g.1.12 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{10}\cdot2$ |
| 56.672.21-56.g.1.15 | $56$ | $2$ | $2$ | $21$ | $10$ | $1^{10}\cdot2$ |
| 56.672.21-56.h.1.6 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{10}\cdot2$ |
| 56.672.21-56.h.1.12 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{10}\cdot2$ |
| 56.672.21-56.h.1.15 | $56$ | $2$ | $2$ | $21$ | $1$ | $1^{10}\cdot2$ |