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Properties

Label 8.96.1.e.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: $32$
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: X447
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.96.1.109

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}1&4\\2&3\end{bmatrix}$, $\begin{bmatrix}7&0\\6&1\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^4$
Contains $-I$: yes
Quadratic refinements: 8.192.1-8.e.2.1, 8.192.1-8.e.2.2, 8.192.1-8.e.2.3, 8.192.1-8.e.2.4, 24.192.1-8.e.2.1, 24.192.1-8.e.2.2, 24.192.1-8.e.2.3, 24.192.1-8.e.2.4, 40.192.1-8.e.2.1, 40.192.1-8.e.2.2, 40.192.1-8.e.2.3, 40.192.1-8.e.2.4, 56.192.1-8.e.2.1, 56.192.1-8.e.2.2, 56.192.1-8.e.2.3, 56.192.1-8.e.2.4, 88.192.1-8.e.2.1, 88.192.1-8.e.2.2, 88.192.1-8.e.2.3, 88.192.1-8.e.2.4, 104.192.1-8.e.2.1, 104.192.1-8.e.2.2, 104.192.1-8.e.2.3, 104.192.1-8.e.2.4, 120.192.1-8.e.2.1, 120.192.1-8.e.2.2, 120.192.1-8.e.2.3, 120.192.1-8.e.2.4, 136.192.1-8.e.2.1, 136.192.1-8.e.2.2, 136.192.1-8.e.2.3, 136.192.1-8.e.2.4, 152.192.1-8.e.2.1, 152.192.1-8.e.2.2, 152.192.1-8.e.2.3, 152.192.1-8.e.2.4, 168.192.1-8.e.2.1, 168.192.1-8.e.2.2, 168.192.1-8.e.2.3, 168.192.1-8.e.2.4, 184.192.1-8.e.2.1, 184.192.1-8.e.2.2, 184.192.1-8.e.2.3, 184.192.1-8.e.2.4, 232.192.1-8.e.2.1, 232.192.1-8.e.2.2, 232.192.1-8.e.2.3, 232.192.1-8.e.2.4, 248.192.1-8.e.2.1, 248.192.1-8.e.2.2, 248.192.1-8.e.2.3, 248.192.1-8.e.2.4, 264.192.1-8.e.2.1, 264.192.1-8.e.2.2, 264.192.1-8.e.2.3, 264.192.1-8.e.2.4, 280.192.1-8.e.2.1, 280.192.1-8.e.2.2, 280.192.1-8.e.2.3, 280.192.1-8.e.2.4, 296.192.1-8.e.2.1, 296.192.1-8.e.2.2, 296.192.1-8.e.2.3, 296.192.1-8.e.2.4, 312.192.1-8.e.2.1, 312.192.1-8.e.2.2, 312.192.1-8.e.2.3, 312.192.1-8.e.2.4, 328.192.1-8.e.2.1, 328.192.1-8.e.2.2, 328.192.1-8.e.2.3, 328.192.1-8.e.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

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
$(0:0:1)$, $(0:1:0)$, $(2:4:1)$, $(2:-4:1)$

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

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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.q.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.48.1.h.2 $8$ $2$ $2$ $1$ $0$ 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.b.3 $8$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
8.192.5.d.2 $8$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.v.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bd.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.17.nf.1 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.ev.1 $24$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
40.192.5.n.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.192.5.v.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.480.33.cl.1 $40$ $5$ $5$ $33$ $7$ $1^{14}\cdot2^{9}$
40.576.33.is.2 $40$ $6$ $6$ $33$ $1$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.lr.1 $40$ $10$ $10$ $65$ $11$ $1^{28}\cdot2^{10}\cdot4^{4}$
56.192.5.n.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.192.5.v.2 $56$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.768.49.ev.2 $56$ $8$ $8$ $49$ $3$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.nj.1 $56$ $21$ $21$ $145$ $19$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.od.1 $56$ $28$ $28$ $193$ $22$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
88.192.5.n.2 $88$ $2$ $2$ $5$ $?$ not computed
88.192.5.v.2 $88$ $2$ $2$ $5$ $?$ not computed
104.192.5.n.2 $104$ $2$ $2$ $5$ $?$ not computed
104.192.5.v.2 $104$ $2$ $2$ $5$ $?$ not computed
120.192.5.eb.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.fh.2 $120$ $2$ $2$ $5$ $?$ not computed
136.192.5.n.2 $136$ $2$ $2$ $5$ $?$ not computed
136.192.5.v.2 $136$ $2$ $2$ $5$ $?$ not computed
152.192.5.n.2 $152$ $2$ $2$ $5$ $?$ not computed
152.192.5.v.2 $152$ $2$ $2$ $5$ $?$ not computed
168.192.5.eb.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fh.2 $168$ $2$ $2$ $5$ $?$ not computed
184.192.5.n.2 $184$ $2$ $2$ $5$ $?$ not computed
184.192.5.v.2 $184$ $2$ $2$ $5$ $?$ not computed
232.192.5.n.2 $232$ $2$ $2$ $5$ $?$ not computed
232.192.5.v.2 $232$ $2$ $2$ $5$ $?$ not computed
248.192.5.n.2 $248$ $2$ $2$ $5$ $?$ not computed
248.192.5.v.2 $248$ $2$ $2$ $5$ $?$ not computed
264.192.5.eb.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.fh.2 $264$ $2$ $2$ $5$ $?$ not computed
280.192.5.dt.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.ez.2 $280$ $2$ $2$ $5$ $?$ not computed
296.192.5.n.2 $296$ $2$ $2$ $5$ $?$ not computed
296.192.5.v.2 $296$ $2$ $2$ $5$ $?$ not computed
312.192.5.eb.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.fh.2 $312$ $2$ $2$ $5$ $?$ not computed
328.192.5.n.2 $328$ $2$ $2$ $5$ $?$ not computed
328.192.5.v.2 $328$ $2$ $2$ $5$ $?$ not computed