↑

Properties

Label 8.96.3.b.1
Level $8$
Index $96$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $64$
Index: $96$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $8^{12}$ Cusp orbits $2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8A3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.96.3.13

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&0\\4&5\end{bmatrix}$, $\begin{bmatrix}3&0\\0&3\end{bmatrix}$, $\begin{bmatrix}3&6\\2&1\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^2\times C_4$
Contains $-I$: yes
Quadratic refinements: 8.192.3-8.b.1.1, 8.192.3-8.b.1.2, 16.192.3-8.b.1.1, 16.192.3-8.b.1.2, 24.192.3-8.b.1.1, 24.192.3-8.b.1.2, 40.192.3-8.b.1.1, 40.192.3-8.b.1.2, 48.192.3-8.b.1.1, 48.192.3-8.b.1.2, 56.192.3-8.b.1.1, 56.192.3-8.b.1.2, 80.192.3-8.b.1.1, 80.192.3-8.b.1.2, 88.192.3-8.b.1.1, 88.192.3-8.b.1.2, 104.192.3-8.b.1.1, 104.192.3-8.b.1.2, 112.192.3-8.b.1.1, 112.192.3-8.b.1.2, 120.192.3-8.b.1.1, 120.192.3-8.b.1.2, 136.192.3-8.b.1.1, 136.192.3-8.b.1.2, 152.192.3-8.b.1.1, 152.192.3-8.b.1.2, 168.192.3-8.b.1.1, 168.192.3-8.b.1.2, 176.192.3-8.b.1.1, 176.192.3-8.b.1.2, 184.192.3-8.b.1.1, 184.192.3-8.b.1.2, 208.192.3-8.b.1.1, 208.192.3-8.b.1.2, 232.192.3-8.b.1.1, 232.192.3-8.b.1.2, 240.192.3-8.b.1.1, 240.192.3-8.b.1.2, 248.192.3-8.b.1.1, 248.192.3-8.b.1.2, 264.192.3-8.b.1.1, 264.192.3-8.b.1.2, 272.192.3-8.b.1.1, 272.192.3-8.b.1.2, 280.192.3-8.b.1.1, 280.192.3-8.b.1.2, 296.192.3-8.b.1.1, 296.192.3-8.b.1.2, 304.192.3-8.b.1.1, 304.192.3-8.b.1.2, 312.192.3-8.b.1.1, 312.192.3-8.b.1.2, 328.192.3-8.b.1.1, 328.192.3-8.b.1.2
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $8$
Full 8-torsion field degree: $16$

Jacobian

Conductor: $2^{16}$
Simple: no
Squarefree: no
Decomposition: $1^{3}$
Newforms: 32.2.a.a$^{2}$, 64.2.a.a

Models

Canonical model in $\mathbb{P}^{ 2 }$

$ 0 $ $=$ $ 4 x^{4} + y^{4} + z^{4} $
 Copy content Toggle raw display

Rational points

This modular curve has no real points, and therefore no rational points.

Maps to other modular curves

$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{(y^{2}-yz+z^{2})^{3}(y^{2}+yz+z^{2})^{3}(y^{4}-y^{2}z^{2}+z^{4})^{3}}{z^{8}y^{8}(y^{4}+z^{4})^{2}}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.1.a.1 $8$ $2$ $2$ $1$ $0$ $1^{2}$
8.48.1.b.1 $8$ $2$ $2$ $1$ $0$ $1^{2}$
8.48.1.bt.1 $8$ $2$ $2$ $1$ $0$ $1^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
16.192.9.b.1 $16$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
16.192.9.b.2 $16$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
24.288.19.b.1 $24$ $3$ $3$ $19$ $3$ $1^{16}$
24.384.21.b.1 $24$ $4$ $4$ $21$ $1$ $1^{18}$
40.480.35.b.1 $40$ $5$ $5$ $35$ $11$ $1^{26}\cdot2^{3}$
40.576.37.b.1 $40$ $6$ $6$ $37$ $4$ $1^{28}\cdot2^{3}$
40.960.69.b.1 $40$ $10$ $10$ $69$ $24$ $1^{54}\cdot2^{6}$
48.192.9.d.1 $48$ $2$ $2$ $9$ $4$ $1^{4}\cdot2$
48.192.9.d.2 $48$ $2$ $2$ $9$ $4$ $1^{4}\cdot2$
56.768.53.b.1 $56$ $8$ $8$ $53$ $12$ $1^{38}\cdot2^{6}$
56.2016.151.b.1 $56$ $21$ $21$ $151$ $55$ $1^{28}\cdot2^{54}\cdot4^{3}$
56.2688.201.b.1 $56$ $28$ $28$ $201$ $67$ $1^{66}\cdot2^{60}\cdot4^{3}$
80.192.9.d.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.d.2 $80$ $2$ $2$ $9$ $?$ not computed
112.192.9.d.1 $112$ $2$ $2$ $9$ $?$ not computed
112.192.9.d.2 $112$ $2$ $2$ $9$ $?$ not computed
176.192.9.d.1 $176$ $2$ $2$ $9$ $?$ not computed
176.192.9.d.2 $176$ $2$ $2$ $9$ $?$ not computed
208.192.9.d.1 $208$ $2$ $2$ $9$ $?$ not computed
208.192.9.d.2 $208$ $2$ $2$ $9$ $?$ not computed
240.192.9.d.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.d.2 $240$ $2$ $2$ $9$ $?$ not computed
272.192.9.d.1 $272$ $2$ $2$ $9$ $?$ not computed
272.192.9.d.2 $272$ $2$ $2$ $9$ $?$ not computed
304.192.9.d.1 $304$ $2$ $2$ $9$ $?$ not computed
304.192.9.d.2 $304$ $2$ $2$ $9$ $?$ not computed