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Properties

Label 8.96.1.f.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 $2^{8}$
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: X449
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.96.1.91

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&0\\4&1\end{bmatrix}$, $\begin{bmatrix}1&0\\4&7\end{bmatrix}$, $\begin{bmatrix}7&0\\4&7\end{bmatrix}$, $\begin{bmatrix}7&4\\0&3\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^4$
Contains $-I$: yes
Quadratic refinements: 8.192.1-8.f.1.1, 8.192.1-8.f.1.2, 8.192.1-8.f.1.3, 8.192.1-8.f.1.4, 8.192.1-8.f.1.5, 8.192.1-8.f.1.6, 16.192.1-8.f.1.1, 16.192.1-8.f.1.2, 16.192.1-8.f.1.3, 16.192.1-8.f.1.4, 24.192.1-8.f.1.1, 24.192.1-8.f.1.2, 24.192.1-8.f.1.3, 24.192.1-8.f.1.4, 24.192.1-8.f.1.5, 24.192.1-8.f.1.6, 40.192.1-8.f.1.1, 40.192.1-8.f.1.2, 40.192.1-8.f.1.3, 40.192.1-8.f.1.4, 40.192.1-8.f.1.5, 40.192.1-8.f.1.6, 48.192.1-8.f.1.1, 48.192.1-8.f.1.2, 48.192.1-8.f.1.3, 48.192.1-8.f.1.4, 56.192.1-8.f.1.1, 56.192.1-8.f.1.2, 56.192.1-8.f.1.3, 56.192.1-8.f.1.4, 56.192.1-8.f.1.5, 56.192.1-8.f.1.6, 80.192.1-8.f.1.1, 80.192.1-8.f.1.2, 80.192.1-8.f.1.3, 80.192.1-8.f.1.4, 88.192.1-8.f.1.1, 88.192.1-8.f.1.2, 88.192.1-8.f.1.3, 88.192.1-8.f.1.4, 88.192.1-8.f.1.5, 88.192.1-8.f.1.6, 104.192.1-8.f.1.1, 104.192.1-8.f.1.2, 104.192.1-8.f.1.3, 104.192.1-8.f.1.4, 104.192.1-8.f.1.5, 104.192.1-8.f.1.6, 112.192.1-8.f.1.1, 112.192.1-8.f.1.2, 112.192.1-8.f.1.3, 112.192.1-8.f.1.4, 120.192.1-8.f.1.1, 120.192.1-8.f.1.2, 120.192.1-8.f.1.3, 120.192.1-8.f.1.4, 120.192.1-8.f.1.5, 120.192.1-8.f.1.6, 136.192.1-8.f.1.1, 136.192.1-8.f.1.2, 136.192.1-8.f.1.3, 136.192.1-8.f.1.4, 136.192.1-8.f.1.5, 136.192.1-8.f.1.6, 152.192.1-8.f.1.1, 152.192.1-8.f.1.2, 152.192.1-8.f.1.3, 152.192.1-8.f.1.4, 152.192.1-8.f.1.5, 152.192.1-8.f.1.6, 168.192.1-8.f.1.1, 168.192.1-8.f.1.2, 168.192.1-8.f.1.3, 168.192.1-8.f.1.4, 168.192.1-8.f.1.5, 168.192.1-8.f.1.6, 176.192.1-8.f.1.1, 176.192.1-8.f.1.2, 176.192.1-8.f.1.3, 176.192.1-8.f.1.4, 184.192.1-8.f.1.1, 184.192.1-8.f.1.2, 184.192.1-8.f.1.3, 184.192.1-8.f.1.4, 184.192.1-8.f.1.5, 184.192.1-8.f.1.6, 208.192.1-8.f.1.1, 208.192.1-8.f.1.2, 208.192.1-8.f.1.3, 208.192.1-8.f.1.4, 232.192.1-8.f.1.1, 232.192.1-8.f.1.2, 232.192.1-8.f.1.3, 232.192.1-8.f.1.4, 232.192.1-8.f.1.5, 232.192.1-8.f.1.6, 240.192.1-8.f.1.1, 240.192.1-8.f.1.2, 240.192.1-8.f.1.3, 240.192.1-8.f.1.4, 248.192.1-8.f.1.1, 248.192.1-8.f.1.2, 248.192.1-8.f.1.3, 248.192.1-8.f.1.4, 248.192.1-8.f.1.5, 248.192.1-8.f.1.6, 264.192.1-8.f.1.1, 264.192.1-8.f.1.2, 264.192.1-8.f.1.3, 264.192.1-8.f.1.4, 264.192.1-8.f.1.5, 264.192.1-8.f.1.6, 272.192.1-8.f.1.1, 272.192.1-8.f.1.2, 272.192.1-8.f.1.3, 272.192.1-8.f.1.4, 280.192.1-8.f.1.1, 280.192.1-8.f.1.2, 280.192.1-8.f.1.3, 280.192.1-8.f.1.4, 280.192.1-8.f.1.5, 280.192.1-8.f.1.6, 296.192.1-8.f.1.1, 296.192.1-8.f.1.2, 296.192.1-8.f.1.3, 296.192.1-8.f.1.4, 296.192.1-8.f.1.5, 296.192.1-8.f.1.6, 304.192.1-8.f.1.1, 304.192.1-8.f.1.2, 304.192.1-8.f.1.3, 304.192.1-8.f.1.4, 312.192.1-8.f.1.1, 312.192.1-8.f.1.2, 312.192.1-8.f.1.3, 312.192.1-8.f.1.4, 312.192.1-8.f.1.5, 312.192.1-8.f.1.6, 328.192.1-8.f.1.1, 328.192.1-8.f.1.2, 328.192.1-8.f.1.3, 328.192.1-8.f.1.4, 328.192.1-8.f.1.5, 328.192.1-8.f.1.6
Cyclic 8-isogeny field degree: $2$
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} - 2 x w - y^{2} $
$=$ $y^{2} + 2 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 2 x^{2} y^{2} - 6 x^{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 between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 2z$
$\displaystyle Z$ $=$ $\displaystyle y$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{(z^{8}-2z^{6}w^{2}+5z^{4}w^{4}-4z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(z-w)^{4}(z+w)^{4}(2z^{2}-w^{2})^{2}}$

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
8.48.0.b.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.0.c.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.0.h.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.0.i.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.1.g.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.m.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.n.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
8.192.5.c.1 $8$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
8.192.5.d.3 $8$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.b.1 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.i.1 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.o.1 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.v.1 $16$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.bh.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.bi.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.17.oz.2 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.fr.2 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
40.192.5.z.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.192.5.ba.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.480.33.ct.2 $40$ $5$ $5$ $33$ $7$ $1^{14}\cdot2^{9}$
40.576.33.jt.2 $40$ $6$ $6$ $33$ $2$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.nl.1 $40$ $10$ $10$ $65$ $12$ $1^{28}\cdot2^{10}\cdot4^{4}$
48.192.5.j.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.bl.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.bs.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.cx.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
56.192.5.z.1 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.ba.1 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.768.49.fr.2 $56$ $8$ $8$ $49$ $6$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.pd.2 $56$ $21$ $21$ $145$ $23$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.px.2 $56$ $28$ $28$ $193$ $29$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
80.192.5.bb.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.cx.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.de.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ev.1 $80$ $2$ $2$ $5$ $?$ not computed
88.192.5.z.1 $88$ $2$ $2$ $5$ $?$ not computed
88.192.5.ba.1 $88$ $2$ $2$ $5$ $?$ not computed
104.192.5.z.1 $104$ $2$ $2$ $5$ $?$ not computed
104.192.5.ba.1 $104$ $2$ $2$ $5$ $?$ not computed
112.192.5.j.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.bl.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.bs.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.cx.2 $112$ $2$ $2$ $5$ $?$ not computed
120.192.5.hp.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hr.1 $120$ $2$ $2$ $5$ $?$ not computed
136.192.5.z.1 $136$ $2$ $2$ $5$ $?$ not computed
136.192.5.ba.1 $136$ $2$ $2$ $5$ $?$ not computed
152.192.5.z.1 $152$ $2$ $2$ $5$ $?$ not computed
152.192.5.ba.1 $152$ $2$ $2$ $5$ $?$ not computed
168.192.5.hp.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.hr.1 $168$ $2$ $2$ $5$ $?$ not computed
176.192.5.j.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.bl.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.bs.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.cx.2 $176$ $2$ $2$ $5$ $?$ not computed
184.192.5.z.1 $184$ $2$ $2$ $5$ $?$ not computed
184.192.5.ba.1 $184$ $2$ $2$ $5$ $?$ not computed
208.192.5.w.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.cx.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.de.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.fa.1 $208$ $2$ $2$ $5$ $?$ not computed
232.192.5.z.1 $232$ $2$ $2$ $5$ $?$ not computed
232.192.5.ba.1 $232$ $2$ $2$ $5$ $?$ not computed
240.192.5.df.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.hx.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ie.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ot.1 $240$ $2$ $2$ $5$ $?$ not computed
248.192.5.z.1 $248$ $2$ $2$ $5$ $?$ not computed
248.192.5.ba.1 $248$ $2$ $2$ $5$ $?$ not computed
264.192.5.hp.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.hr.1 $264$ $2$ $2$ $5$ $?$ not computed
272.192.5.j.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.cx.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.de.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.fn.1 $272$ $2$ $2$ $5$ $?$ not computed
280.192.5.hh.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.hj.1 $280$ $2$ $2$ $5$ $?$ not computed
296.192.5.z.1 $296$ $2$ $2$ $5$ $?$ not computed
296.192.5.ba.1 $296$ $2$ $2$ $5$ $?$ not computed
304.192.5.j.1 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.bl.1 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.bs.1 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.cx.2 $304$ $2$ $2$ $5$ $?$ not computed
312.192.5.hp.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.hr.1 $312$ $2$ $2$ $5$ $?$ not computed
328.192.5.z.1 $328$ $2$ $2$ $5$ $?$ not computed
328.192.5.ba.1 $328$ $2$ $2$ $5$ $?$ not computed