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Properties

Label 8.96.1.h.2
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: $32$
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 $2^{6}\cdot4$
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: X455
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.96.1.154

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&0\\0&3\end{bmatrix}$, $\begin{bmatrix}5&2\\4&3\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.192.1-8.h.2.1, 8.192.1-8.h.2.2, 8.192.1-8.h.2.3, 8.192.1-8.h.2.4, 16.192.1-8.h.2.1, 16.192.1-8.h.2.2, 16.192.1-8.h.2.3, 16.192.1-8.h.2.4, 16.192.1-8.h.2.5, 16.192.1-8.h.2.6, 24.192.1-8.h.2.1, 24.192.1-8.h.2.2, 24.192.1-8.h.2.3, 24.192.1-8.h.2.4, 40.192.1-8.h.2.1, 40.192.1-8.h.2.2, 40.192.1-8.h.2.3, 40.192.1-8.h.2.4, 48.192.1-8.h.2.1, 48.192.1-8.h.2.2, 48.192.1-8.h.2.3, 48.192.1-8.h.2.4, 48.192.1-8.h.2.5, 48.192.1-8.h.2.6, 56.192.1-8.h.2.1, 56.192.1-8.h.2.2, 56.192.1-8.h.2.3, 56.192.1-8.h.2.4, 80.192.1-8.h.2.1, 80.192.1-8.h.2.2, 80.192.1-8.h.2.3, 80.192.1-8.h.2.4, 80.192.1-8.h.2.5, 80.192.1-8.h.2.6, 88.192.1-8.h.2.1, 88.192.1-8.h.2.2, 88.192.1-8.h.2.3, 88.192.1-8.h.2.4, 104.192.1-8.h.2.1, 104.192.1-8.h.2.2, 104.192.1-8.h.2.3, 104.192.1-8.h.2.4, 112.192.1-8.h.2.1, 112.192.1-8.h.2.2, 112.192.1-8.h.2.3, 112.192.1-8.h.2.4, 112.192.1-8.h.2.5, 112.192.1-8.h.2.6, 120.192.1-8.h.2.1, 120.192.1-8.h.2.2, 120.192.1-8.h.2.3, 120.192.1-8.h.2.4, 136.192.1-8.h.2.1, 136.192.1-8.h.2.2, 136.192.1-8.h.2.3, 136.192.1-8.h.2.4, 152.192.1-8.h.2.1, 152.192.1-8.h.2.2, 152.192.1-8.h.2.3, 152.192.1-8.h.2.4, 168.192.1-8.h.2.1, 168.192.1-8.h.2.2, 168.192.1-8.h.2.3, 168.192.1-8.h.2.4, 176.192.1-8.h.2.1, 176.192.1-8.h.2.2, 176.192.1-8.h.2.3, 176.192.1-8.h.2.4, 176.192.1-8.h.2.5, 176.192.1-8.h.2.6, 184.192.1-8.h.2.1, 184.192.1-8.h.2.2, 184.192.1-8.h.2.3, 184.192.1-8.h.2.4, 208.192.1-8.h.2.1, 208.192.1-8.h.2.2, 208.192.1-8.h.2.3, 208.192.1-8.h.2.4, 208.192.1-8.h.2.5, 208.192.1-8.h.2.6, 232.192.1-8.h.2.1, 232.192.1-8.h.2.2, 232.192.1-8.h.2.3, 232.192.1-8.h.2.4, 240.192.1-8.h.2.1, 240.192.1-8.h.2.2, 240.192.1-8.h.2.3, 240.192.1-8.h.2.4, 240.192.1-8.h.2.5, 240.192.1-8.h.2.6, 248.192.1-8.h.2.1, 248.192.1-8.h.2.2, 248.192.1-8.h.2.3, 248.192.1-8.h.2.4, 264.192.1-8.h.2.1, 264.192.1-8.h.2.2, 264.192.1-8.h.2.3, 264.192.1-8.h.2.4, 272.192.1-8.h.2.1, 272.192.1-8.h.2.2, 272.192.1-8.h.2.3, 272.192.1-8.h.2.4, 272.192.1-8.h.2.5, 272.192.1-8.h.2.6, 280.192.1-8.h.2.1, 280.192.1-8.h.2.2, 280.192.1-8.h.2.3, 280.192.1-8.h.2.4, 296.192.1-8.h.2.1, 296.192.1-8.h.2.2, 296.192.1-8.h.2.3, 296.192.1-8.h.2.4, 304.192.1-8.h.2.1, 304.192.1-8.h.2.2, 304.192.1-8.h.2.3, 304.192.1-8.h.2.4, 304.192.1-8.h.2.5, 304.192.1-8.h.2.6, 312.192.1-8.h.2.1, 312.192.1-8.h.2.2, 312.192.1-8.h.2.3, 312.192.1-8.h.2.4, 328.192.1-8.h.2.1, 328.192.1-8.h.2.2, 328.192.1-8.h.2.3, 328.192.1-8.h.2.4
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $4$
Full 8-torsion field degree: $16$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

$ 0 $ $=$ $ 2 x^{2} - 2 z^{2} - w^{2} $
$=$ $x^{2} - 2 x w - 2 y^{2} + 2 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 2 x^{2} y^{2} - 12 x^{2} z^{2} + 4 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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^2\,\frac{11524964352xz^{22}w+131125133312xz^{20}w^{3}+559345844224xz^{18}w^{5}+1316776366080xz^{16}w^{7}+1993853575168xz^{14}w^{9}+2089570924544xz^{12}w^{11}+1568560736256xz^{10}w^{13}+851139791360xz^{8}w^{15}+329339305216xz^{6}w^{17}+87005395392xz^{4}w^{19}+14190810432xz^{2}w^{21}+1086679440xw^{23}-1593413632z^{24}-46383316992z^{22}w^{2}-307595692032z^{20}w^{4}-985761542144z^{18}w^{6}-1921750957824z^{16}w^{8}-2530468251648z^{14}w^{10}-2371525650176z^{12}w^{12}-1619709513216z^{10}w^{14}-808500681840z^{8}w^{16}-289767151168z^{6}w^{18}-71172324168z^{4}w^{20}-10802816688z^{2}w^{22}-768398401w^{24}}{z^{4}(2z^{2}+w^{2})^{2}(1050624xz^{14}w+26377728xz^{12}w^{3}+180662272xz^{10}w^{5}+523172992xz^{8}w^{7}+754244736xz^{6}w^{9}+569087776xz^{4}w^{11}+214828480xz^{2}w^{13}+31988856xw^{15}-82944z^{16}-6538752z^{14}w^{2}-79822144z^{12}w^{4}-357074240z^{10}w^{6}-763883952z^{8}w^{8}-871093824z^{6}w^{10}-543002732z^{4}w^{12}-174526212z^{2}w^{14}-22619537w^{16})}$

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.e.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.0.f.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.i.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.j.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.k.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.c.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.j.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.p.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.w.2 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.9.bt.1 $16$ $2$ $2$ $9$ $0$ $1^{4}\cdot2^{2}$
16.192.9.bw.1 $16$ $2$ $2$ $9$ $2$ $1^{4}\cdot2^{2}$
24.288.17.pn.1 $24$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
24.384.17.gb.1 $24$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
40.480.33.cz.1 $40$ $5$ $5$ $33$ $7$ $1^{14}\cdot2^{9}$
40.576.33.kd.1 $40$ $6$ $6$ $33$ $5$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.nz.1 $40$ $10$ $10$ $65$ $13$ $1^{28}\cdot2^{10}\cdot4^{4}$
48.192.5.k.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.bm.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.bt.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.cy.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.9.gj.1 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot2^{2}$
48.192.9.go.1 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot2^{2}$
56.768.49.gb.1 $56$ $8$ $8$ $49$ $5$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.pr.1 $56$ $21$ $21$ $145$ $27$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.ql.1 $56$ $28$ $28$ $193$ $32$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
80.192.5.bc.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.cy.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.df.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ew.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.9.jx.1 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.kc.1 $80$ $2$ $2$ $9$ $?$ not computed
112.192.5.k.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.bm.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.bt.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.cy.2 $112$ $2$ $2$ $5$ $?$ not computed
112.192.9.gj.1 $112$ $2$ $2$ $9$ $?$ not computed
112.192.9.go.1 $112$ $2$ $2$ $9$ $?$ not computed
176.192.5.k.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.bm.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.bt.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.cy.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.9.gj.1 $176$ $2$ $2$ $9$ $?$ not computed
176.192.9.go.1 $176$ $2$ $2$ $9$ $?$ not computed
208.192.5.x.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.cy.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.df.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.fb.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.9.jx.1 $208$ $2$ $2$ $9$ $?$ not computed
208.192.9.kc.1 $208$ $2$ $2$ $9$ $?$ not computed
240.192.5.dg.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.hy.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.if.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ou.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.bkb.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bkm.1 $240$ $2$ $2$ $9$ $?$ not computed
272.192.5.k.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.cy.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.df.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.fo.2 $272$ $2$ $2$ $5$ $?$ not computed
272.192.9.jx.1 $272$ $2$ $2$ $9$ $?$ not computed
272.192.9.kc.1 $272$ $2$ $2$ $9$ $?$ not computed
304.192.5.k.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.bm.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.bt.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.cy.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.9.gj.1 $304$ $2$ $2$ $9$ $?$ not computed
304.192.9.go.1 $304$ $2$ $2$ $9$ $?$ not computed