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Properties

Label 8.96.1.i.1
Level $8$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $64$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse and Zureick-Brown (RZB) label: X442
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.96.1.173

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&6\\0&5\end{bmatrix}$, $\begin{bmatrix}5&0\\0&3\end{bmatrix}$, $\begin{bmatrix}7&2\\0&3\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.192.1-8.i.1.1, 8.192.1-8.i.1.2, 8.192.1-8.i.1.3, 8.192.1-8.i.1.4, 16.192.1-8.i.1.1, 16.192.1-8.i.1.2, 24.192.1-8.i.1.1, 24.192.1-8.i.1.2, 24.192.1-8.i.1.3, 24.192.1-8.i.1.4, 40.192.1-8.i.1.1, 40.192.1-8.i.1.2, 40.192.1-8.i.1.3, 40.192.1-8.i.1.4, 48.192.1-8.i.1.1, 48.192.1-8.i.1.2, 56.192.1-8.i.1.1, 56.192.1-8.i.1.2, 56.192.1-8.i.1.3, 56.192.1-8.i.1.4, 80.192.1-8.i.1.1, 80.192.1-8.i.1.2, 88.192.1-8.i.1.1, 88.192.1-8.i.1.2, 88.192.1-8.i.1.3, 88.192.1-8.i.1.4, 104.192.1-8.i.1.1, 104.192.1-8.i.1.2, 104.192.1-8.i.1.3, 104.192.1-8.i.1.4, 112.192.1-8.i.1.1, 112.192.1-8.i.1.2, 120.192.1-8.i.1.1, 120.192.1-8.i.1.2, 120.192.1-8.i.1.3, 120.192.1-8.i.1.4, 136.192.1-8.i.1.1, 136.192.1-8.i.1.2, 136.192.1-8.i.1.3, 136.192.1-8.i.1.4, 152.192.1-8.i.1.1, 152.192.1-8.i.1.2, 152.192.1-8.i.1.3, 152.192.1-8.i.1.4, 168.192.1-8.i.1.1, 168.192.1-8.i.1.2, 168.192.1-8.i.1.3, 168.192.1-8.i.1.4, 176.192.1-8.i.1.1, 176.192.1-8.i.1.2, 184.192.1-8.i.1.1, 184.192.1-8.i.1.2, 184.192.1-8.i.1.3, 184.192.1-8.i.1.4, 208.192.1-8.i.1.1, 208.192.1-8.i.1.2, 232.192.1-8.i.1.1, 232.192.1-8.i.1.2, 232.192.1-8.i.1.3, 232.192.1-8.i.1.4, 240.192.1-8.i.1.1, 240.192.1-8.i.1.2, 248.192.1-8.i.1.1, 248.192.1-8.i.1.2, 248.192.1-8.i.1.3, 248.192.1-8.i.1.4, 264.192.1-8.i.1.1, 264.192.1-8.i.1.2, 264.192.1-8.i.1.3, 264.192.1-8.i.1.4, 272.192.1-8.i.1.1, 272.192.1-8.i.1.2, 280.192.1-8.i.1.1, 280.192.1-8.i.1.2, 280.192.1-8.i.1.3, 280.192.1-8.i.1.4, 296.192.1-8.i.1.1, 296.192.1-8.i.1.2, 296.192.1-8.i.1.3, 296.192.1-8.i.1.4, 304.192.1-8.i.1.1, 304.192.1-8.i.1.2, 312.192.1-8.i.1.1, 312.192.1-8.i.1.2, 312.192.1-8.i.1.3, 312.192.1-8.i.1.4, 328.192.1-8.i.1.1, 328.192.1-8.i.1.2, 328.192.1-8.i.1.3, 328.192.1-8.i.1.4
Cyclic 8-isogeny field degree: $1$
Cyclic 8-torsion field degree: $4$
Full 8-torsion field degree: $16$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ x^{2} + y^{2} + z^{2} $
$=$ $2 y^{2} - 2 z^{2} + w^{2}$
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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 embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{(16z^{8}-16z^{6}w^{2}+20z^{4}w^{4}-8z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(2z-w)^{2}(2z+w)^{2}(2z^{2}-w^{2})^{4}}$

Modular covers

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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
8.48.0.d.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.0.e.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.0.j.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.0.k.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.1.l.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.m.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.p.1 $8$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
16.192.5.g.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.s.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.17.tg.2 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.ht.2 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
40.480.33.dq.2 $40$ $5$ $5$ $33$ $11$ $1^{14}\cdot2^{9}$
40.576.33.mi.2 $40$ $6$ $6$ $33$ $2$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.rs.2 $40$ $10$ $10$ $65$ $16$ $1^{28}\cdot2^{10}\cdot4^{4}$
48.192.5.bu.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.192.5.ce.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
56.768.49.ht.2 $56$ $8$ $8$ $49$ $6$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.tk.2 $56$ $21$ $21$ $145$ $23$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.ue.2 $56$ $28$ $28$ $193$ $29$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
80.192.5.dg.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.du.1 $80$ $2$ $2$ $5$ $?$ not computed
112.192.5.bu.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.ce.1 $112$ $2$ $2$ $5$ $?$ not computed
176.192.5.bu.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.ce.1 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.dg.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.du.1 $208$ $2$ $2$ $5$ $?$ not computed
240.192.5.jy.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.lc.1 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.dg.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.du.1 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.bu.1 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.ce.1 $304$ $2$ $2$ $5$ $?$ not computed