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Properties

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

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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 $2^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3$
$\overline{\Q}$-gonality: $3$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8A3
Rouse and Zureick-Brown (RZB) label: X543
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.96.3.14

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}0&1\\1&0\end{bmatrix}$, $\begin{bmatrix}0&1\\5&0\end{bmatrix}$, $\begin{bmatrix}0&3\\7&0\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.192.3-8.n.1.1, 8.192.3-8.n.1.2, 16.192.3-8.n.1.1, 16.192.3-8.n.1.2, 24.192.3-8.n.1.1, 24.192.3-8.n.1.2, 40.192.3-8.n.1.1, 40.192.3-8.n.1.2, 48.192.3-8.n.1.1, 48.192.3-8.n.1.2, 56.192.3-8.n.1.1, 56.192.3-8.n.1.2, 80.192.3-8.n.1.1, 80.192.3-8.n.1.2, 88.192.3-8.n.1.1, 88.192.3-8.n.1.2, 104.192.3-8.n.1.1, 104.192.3-8.n.1.2, 112.192.3-8.n.1.1, 112.192.3-8.n.1.2, 120.192.3-8.n.1.1, 120.192.3-8.n.1.2, 136.192.3-8.n.1.1, 136.192.3-8.n.1.2, 152.192.3-8.n.1.1, 152.192.3-8.n.1.2, 168.192.3-8.n.1.1, 168.192.3-8.n.1.2, 176.192.3-8.n.1.1, 176.192.3-8.n.1.2, 184.192.3-8.n.1.1, 184.192.3-8.n.1.2, 208.192.3-8.n.1.1, 208.192.3-8.n.1.2, 232.192.3-8.n.1.1, 232.192.3-8.n.1.2, 240.192.3-8.n.1.1, 240.192.3-8.n.1.2, 248.192.3-8.n.1.1, 248.192.3-8.n.1.2, 264.192.3-8.n.1.1, 264.192.3-8.n.1.2, 272.192.3-8.n.1.1, 272.192.3-8.n.1.2, 280.192.3-8.n.1.1, 280.192.3-8.n.1.2, 296.192.3-8.n.1.1, 296.192.3-8.n.1.2, 304.192.3-8.n.1.1, 304.192.3-8.n.1.2, 312.192.3-8.n.1.1, 312.192.3-8.n.1.2, 328.192.3-8.n.1.1, 328.192.3-8.n.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 $ $=$ $ 2 x^{4} + y^{4} - 6 y^{2} z^{2} + z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 $X(1)$ :

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

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
8.48.1.bm.1 $8$ $2$ $2$ $1$ $0$ $1^{2}$
8.48.1.bt.1 $8$ $2$ $2$ $1$ $0$ $1^{2}$
$X_{\mathrm{sp}}^+(8)$ $8$ $2$ $2$ $1$ $0$ $1^{2}$

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.dl.1 $16$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
16.192.9.dp.1 $16$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
16.192.9.ff.1 $16$ $2$ $2$ $9$ $3$ $1^{4}\cdot2$
16.192.9.fj.1 $16$ $2$ $2$ $9$ $3$ $1^{4}\cdot2$
24.288.19.fy.1 $24$ $3$ $3$ $19$ $4$ $1^{16}$
24.384.21.ef.1 $24$ $4$ $4$ $21$ $1$ $1^{18}$
40.480.35.cn.1 $40$ $5$ $5$ $35$ $12$ $1^{26}\cdot2^{3}$
40.576.37.jh.1 $40$ $6$ $6$ $37$ $4$ $1^{28}\cdot2^{3}$
40.960.69.kh.1 $40$ $10$ $10$ $69$ $26$ $1^{54}\cdot2^{6}$
48.192.9.qf.1 $48$ $2$ $2$ $9$ $5$ $1^{4}\cdot2$
48.192.9.qn.1 $48$ $2$ $2$ $9$ $5$ $1^{4}\cdot2$
48.192.9.sy.1 $48$ $2$ $2$ $9$ $3$ $1^{4}\cdot2$
48.192.9.tg.1 $48$ $2$ $2$ $9$ $3$ $1^{4}\cdot2$
56.768.53.ef.1 $56$ $8$ $8$ $53$ $11$ $1^{38}\cdot2^{6}$
56.2016.151.fy.1 $56$ $21$ $21$ $151$ $61$ $1^{28}\cdot2^{54}\cdot4^{3}$
56.2688.201.fz.1 $56$ $28$ $28$ $201$ $72$ $1^{66}\cdot2^{60}\cdot4^{3}$
80.192.9.vi.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.vq.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.yf.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.yn.1 $80$ $2$ $2$ $9$ $?$ not computed
112.192.9.nj.1 $112$ $2$ $2$ $9$ $?$ not computed
112.192.9.nr.1 $112$ $2$ $2$ $9$ $?$ not computed
112.192.9.pq.1 $112$ $2$ $2$ $9$ $?$ not computed
112.192.9.py.1 $112$ $2$ $2$ $9$ $?$ not computed
176.192.9.ni.1 $176$ $2$ $2$ $9$ $?$ not computed
176.192.9.nq.1 $176$ $2$ $2$ $9$ $?$ not computed
176.192.9.pp.1 $176$ $2$ $2$ $9$ $?$ not computed
176.192.9.px.1 $176$ $2$ $2$ $9$ $?$ not computed
208.192.9.vi.1 $208$ $2$ $2$ $9$ $?$ not computed
208.192.9.vq.1 $208$ $2$ $2$ $9$ $?$ not computed
208.192.9.yf.1 $208$ $2$ $2$ $9$ $?$ not computed
208.192.9.yn.1 $208$ $2$ $2$ $9$ $?$ not computed
240.192.9.dat.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.dbj.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.die.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.diu.1 $240$ $2$ $2$ $9$ $?$ not computed
272.192.9.us.1 $272$ $2$ $2$ $9$ $?$ not computed
272.192.9.va.1 $272$ $2$ $2$ $9$ $?$ not computed
272.192.9.yf.1 $272$ $2$ $2$ $9$ $?$ not computed
272.192.9.yn.1 $272$ $2$ $2$ $9$ $?$ not computed
304.192.9.ni.1 $304$ $2$ $2$ $9$ $?$ not computed
304.192.9.nq.1 $304$ $2$ $2$ $9$ $?$ not computed
304.192.9.pp.1 $304$ $2$ $2$ $9$ $?$ not computed
304.192.9.px.1 $304$ $2$ $2$ $9$ $?$ not computed