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Properties

Label 8.96.1.f.2
Level $8$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $4$

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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: $16$ (of which $4$ are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $1^{4}\cdot2^{2}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse and Zureick-Brown (RZB) label: X450
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.96.1.96

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$, $\begin{bmatrix}7&4\\4&1\end{bmatrix}$, $\begin{bmatrix}7&4\\4&3\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^4$
Contains $-I$: yes
Quadratic refinements: 8.192.1-8.f.2.1, 8.192.1-8.f.2.2, 8.192.1-8.f.2.3, 8.192.1-8.f.2.4, 8.192.1-8.f.2.5, 8.192.1-8.f.2.6, 16.192.1-8.f.2.1, 16.192.1-8.f.2.2, 16.192.1-8.f.2.3, 16.192.1-8.f.2.4, 24.192.1-8.f.2.1, 24.192.1-8.f.2.2, 24.192.1-8.f.2.3, 24.192.1-8.f.2.4, 24.192.1-8.f.2.5, 24.192.1-8.f.2.6, 40.192.1-8.f.2.1, 40.192.1-8.f.2.2, 40.192.1-8.f.2.3, 40.192.1-8.f.2.4, 40.192.1-8.f.2.5, 40.192.1-8.f.2.6, 48.192.1-8.f.2.1, 48.192.1-8.f.2.2, 48.192.1-8.f.2.3, 48.192.1-8.f.2.4, 56.192.1-8.f.2.1, 56.192.1-8.f.2.2, 56.192.1-8.f.2.3, 56.192.1-8.f.2.4, 56.192.1-8.f.2.5, 56.192.1-8.f.2.6, 80.192.1-8.f.2.1, 80.192.1-8.f.2.2, 80.192.1-8.f.2.3, 80.192.1-8.f.2.4, 88.192.1-8.f.2.1, 88.192.1-8.f.2.2, 88.192.1-8.f.2.3, 88.192.1-8.f.2.4, 88.192.1-8.f.2.5, 88.192.1-8.f.2.6, 104.192.1-8.f.2.1, 104.192.1-8.f.2.2, 104.192.1-8.f.2.3, 104.192.1-8.f.2.4, 104.192.1-8.f.2.5, 104.192.1-8.f.2.6, 112.192.1-8.f.2.1, 112.192.1-8.f.2.2, 112.192.1-8.f.2.3, 112.192.1-8.f.2.4, 120.192.1-8.f.2.1, 120.192.1-8.f.2.2, 120.192.1-8.f.2.3, 120.192.1-8.f.2.4, 120.192.1-8.f.2.5, 120.192.1-8.f.2.6, 136.192.1-8.f.2.1, 136.192.1-8.f.2.2, 136.192.1-8.f.2.3, 136.192.1-8.f.2.4, 136.192.1-8.f.2.5, 136.192.1-8.f.2.6, 152.192.1-8.f.2.1, 152.192.1-8.f.2.2, 152.192.1-8.f.2.3, 152.192.1-8.f.2.4, 152.192.1-8.f.2.5, 152.192.1-8.f.2.6, 168.192.1-8.f.2.1, 168.192.1-8.f.2.2, 168.192.1-8.f.2.3, 168.192.1-8.f.2.4, 168.192.1-8.f.2.5, 168.192.1-8.f.2.6, 176.192.1-8.f.2.1, 176.192.1-8.f.2.2, 176.192.1-8.f.2.3, 176.192.1-8.f.2.4, 184.192.1-8.f.2.1, 184.192.1-8.f.2.2, 184.192.1-8.f.2.3, 184.192.1-8.f.2.4, 184.192.1-8.f.2.5, 184.192.1-8.f.2.6, 208.192.1-8.f.2.1, 208.192.1-8.f.2.2, 208.192.1-8.f.2.3, 208.192.1-8.f.2.4, 232.192.1-8.f.2.1, 232.192.1-8.f.2.2, 232.192.1-8.f.2.3, 232.192.1-8.f.2.4, 232.192.1-8.f.2.5, 232.192.1-8.f.2.6, 240.192.1-8.f.2.1, 240.192.1-8.f.2.2, 240.192.1-8.f.2.3, 240.192.1-8.f.2.4, 248.192.1-8.f.2.1, 248.192.1-8.f.2.2, 248.192.1-8.f.2.3, 248.192.1-8.f.2.4, 248.192.1-8.f.2.5, 248.192.1-8.f.2.6, 264.192.1-8.f.2.1, 264.192.1-8.f.2.2, 264.192.1-8.f.2.3, 264.192.1-8.f.2.4, 264.192.1-8.f.2.5, 264.192.1-8.f.2.6, 272.192.1-8.f.2.1, 272.192.1-8.f.2.2, 272.192.1-8.f.2.3, 272.192.1-8.f.2.4, 280.192.1-8.f.2.1, 280.192.1-8.f.2.2, 280.192.1-8.f.2.3, 280.192.1-8.f.2.4, 280.192.1-8.f.2.5, 280.192.1-8.f.2.6, 296.192.1-8.f.2.1, 296.192.1-8.f.2.2, 296.192.1-8.f.2.3, 296.192.1-8.f.2.4, 296.192.1-8.f.2.5, 296.192.1-8.f.2.6, 304.192.1-8.f.2.1, 304.192.1-8.f.2.2, 304.192.1-8.f.2.3, 304.192.1-8.f.2.4, 312.192.1-8.f.2.1, 312.192.1-8.f.2.2, 312.192.1-8.f.2.3, 312.192.1-8.f.2.4, 312.192.1-8.f.2.5, 312.192.1-8.f.2.6, 328.192.1-8.f.2.1, 328.192.1-8.f.2.2, 328.192.1-8.f.2.3, 328.192.1-8.f.2.4, 328.192.1-8.f.2.5, 328.192.1-8.f.2.6
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

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 4x $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:0:1)$, $(0:1:0)$, $(2:0:1)$, $(-2:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^4}\cdot\frac{11328x^{2}y^{28}z^{2}-489338880x^{2}y^{24}z^{6}+2591686066176x^{2}y^{20}z^{10}+1085247282216960x^{2}y^{16}z^{14}+303432552482340864x^{2}y^{12}z^{18}+37073617649884200960x^{2}y^{8}z^{22}+830103506406674006016x^{2}y^{4}z^{26}+1180591550348667125760x^{2}z^{30}-32xy^{30}z+41366016xy^{26}z^{5}-10544873472xy^{22}z^{9}+168210524536832xy^{18}z^{13}+68116365617135616xy^{14}z^{17}+10403318682573864960xy^{10}z^{21}+553402316713728409600xy^{6}z^{25}+3246626974565067128832xy^{2}z^{29}+y^{32}-265728y^{28}z^{4}+51360677888y^{24}z^{8}+20046973239296y^{20}z^{12}+7673506078654464y^{16}z^{16}+1292522115118923776y^{12}z^{20}+96845465623164092416y^{8}z^{24}+737869666191358820352y^{4}z^{28}+281474976710656z^{32}}{z^{2}y^{8}(x^{2}y^{20}+10048x^{2}y^{16}z^{4}-68075520x^{2}y^{12}z^{8}+81606737920x^{2}y^{8}z^{12}+16492892520448x^{2}y^{4}z^{16}+70367670435840x^{2}z^{20}-784xy^{18}z^{3}+204800xy^{14}z^{7}+1072496640xy^{10}z^{11}+5497507807232xy^{6}z^{15}+123145570746368xy^{2}z^{19}-24y^{20}z^{2}+312832y^{16}z^{6}-532545536y^{12}z^{10}+687196864512y^{8}z^{14}+26387339542528y^{4}z^{18}+4294967296z^{22})}$

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