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Properties

Label 8.48.1.bg.1
Level $8$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $32$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
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: 8F1
Rouse and Zureick-Brown (RZB) label: X261
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.90

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&3\\0&3\end{bmatrix}$, $\begin{bmatrix}1&7\\6&7\end{bmatrix}$, $\begin{bmatrix}5&4\\0&5\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^3:C_4$
Contains $-I$: yes
Quadratic refinements: 8.96.1-8.bg.1.1, 16.96.1-8.bg.1.1, 16.96.1-8.bg.1.2, 16.96.1-8.bg.1.3, 16.96.1-8.bg.1.4, 24.96.1-8.bg.1.1, 40.96.1-8.bg.1.1, 48.96.1-8.bg.1.1, 48.96.1-8.bg.1.2, 48.96.1-8.bg.1.3, 48.96.1-8.bg.1.4, 56.96.1-8.bg.1.1, 80.96.1-8.bg.1.1, 80.96.1-8.bg.1.2, 80.96.1-8.bg.1.3, 80.96.1-8.bg.1.4, 88.96.1-8.bg.1.1, 104.96.1-8.bg.1.1, 112.96.1-8.bg.1.1, 112.96.1-8.bg.1.2, 112.96.1-8.bg.1.3, 112.96.1-8.bg.1.4, 120.96.1-8.bg.1.1, 136.96.1-8.bg.1.1, 152.96.1-8.bg.1.1, 168.96.1-8.bg.1.1, 176.96.1-8.bg.1.1, 176.96.1-8.bg.1.2, 176.96.1-8.bg.1.3, 176.96.1-8.bg.1.4, 184.96.1-8.bg.1.1, 208.96.1-8.bg.1.1, 208.96.1-8.bg.1.2, 208.96.1-8.bg.1.3, 208.96.1-8.bg.1.4, 232.96.1-8.bg.1.1, 240.96.1-8.bg.1.1, 240.96.1-8.bg.1.2, 240.96.1-8.bg.1.3, 240.96.1-8.bg.1.4, 248.96.1-8.bg.1.1, 264.96.1-8.bg.1.1, 272.96.1-8.bg.1.1, 272.96.1-8.bg.1.2, 272.96.1-8.bg.1.3, 272.96.1-8.bg.1.4, 280.96.1-8.bg.1.1, 296.96.1-8.bg.1.1, 304.96.1-8.bg.1.1, 304.96.1-8.bg.1.2, 304.96.1-8.bg.1.3, 304.96.1-8.bg.1.4, 312.96.1-8.bg.1.1, 328.96.1-8.bg.1.1
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $8$
Full 8-torsion field degree: $32$

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} + z w $
$=$ $2 x^{2} + 8 x y - y^{2} + 2 z^{2} - 3 z w + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 4 x^{2} z^{2} + 8 x y z^{2} - 2 y^{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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

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

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
8.24.0.s.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.v.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.bh.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.bi.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.1.p.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.u.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.x.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
16.96.3.da.1 $16$ $2$ $2$ $3$ $1$ $1^{2}$
16.96.3.db.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.dc.1 $16$ $2$ $2$ $3$ $1$ $1^{2}$
16.96.5.bw.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.96.5.by.1 $16$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.144.9.vr.1 $24$ $3$ $3$ $9$ $3$ $1^{8}$
24.192.9.ij.1 $24$ $4$ $4$ $9$ $1$ $1^{8}$
40.240.17.fp.1 $40$ $5$ $5$ $17$ $6$ $1^{14}\cdot2$
40.288.17.nk.1 $40$ $6$ $6$ $17$ $3$ $1^{14}\cdot2$
40.480.33.yd.1 $40$ $10$ $10$ $33$ $13$ $1^{28}\cdot2^{2}$
48.96.3.go.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.gp.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.gq.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.5.ee.1 $48$ $2$ $2$ $5$ $4$ $1^{2}\cdot2$
48.96.5.eg.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
56.384.25.ij.1 $56$ $8$ $8$ $25$ $6$ $1^{20}\cdot2^{2}$
56.1008.73.vr.1 $56$ $21$ $21$ $73$ $32$ $1^{16}\cdot2^{26}\cdot4$
56.1344.97.vj.1 $56$ $28$ $28$ $97$ $38$ $1^{36}\cdot2^{28}\cdot4$
80.96.3.hq.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.hr.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.hs.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.5.ei.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ek.1 $80$ $2$ $2$ $5$ $?$ not computed
112.96.3.go.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.gp.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.gq.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.5.ee.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.eg.1 $112$ $2$ $2$ $5$ $?$ not computed
176.96.3.go.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.gp.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.gq.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.5.ee.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.eg.1 $176$ $2$ $2$ $5$ $?$ not computed
208.96.3.hq.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.hr.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.hs.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.5.ei.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.ek.1 $208$ $2$ $2$ $5$ $?$ not computed
240.96.3.tm.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.tn.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.to.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.5.lg.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.li.1 $240$ $2$ $2$ $5$ $?$ not computed
272.96.3.hq.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.hr.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.hs.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.5.ei.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.ek.1 $272$ $2$ $2$ $5$ $?$ not computed
304.96.3.go.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.gp.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.gq.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.5.ee.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.eg.1 $304$ $2$ $2$ $5$ $?$ not computed