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Properties

Label 8.48.1.g.1
Level $8$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $64$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse and Zureick-Brown (RZB) label: X248
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.18

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}3&4\\4&7\end{bmatrix}$, $\begin{bmatrix}5&4\\0&5\end{bmatrix}$, $\begin{bmatrix}7&0\\4&5\end{bmatrix}$, $\begin{bmatrix}7&4\\0&1\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^5$
Contains $-I$: yes
Quadratic refinements: 8.96.1-8.g.1.1, 8.96.1-8.g.1.2, 8.96.1-8.g.1.3, 8.96.1-8.g.1.4, 8.96.1-8.g.1.5, 8.96.1-8.g.1.6, 8.96.1-8.g.1.7, 8.96.1-8.g.1.8, 8.96.1-8.g.1.9, 8.96.1-8.g.1.10, 8.96.1-8.g.1.11, 8.96.1-8.g.1.12, 24.96.1-8.g.1.1, 24.96.1-8.g.1.2, 24.96.1-8.g.1.3, 24.96.1-8.g.1.4, 24.96.1-8.g.1.5, 24.96.1-8.g.1.6, 24.96.1-8.g.1.7, 24.96.1-8.g.1.8, 24.96.1-8.g.1.9, 24.96.1-8.g.1.10, 24.96.1-8.g.1.11, 24.96.1-8.g.1.12, 40.96.1-8.g.1.1, 40.96.1-8.g.1.2, 40.96.1-8.g.1.3, 40.96.1-8.g.1.4, 40.96.1-8.g.1.5, 40.96.1-8.g.1.6, 40.96.1-8.g.1.7, 40.96.1-8.g.1.8, 40.96.1-8.g.1.9, 40.96.1-8.g.1.10, 40.96.1-8.g.1.11, 40.96.1-8.g.1.12, 56.96.1-8.g.1.1, 56.96.1-8.g.1.2, 56.96.1-8.g.1.3, 56.96.1-8.g.1.4, 56.96.1-8.g.1.5, 56.96.1-8.g.1.6, 56.96.1-8.g.1.7, 56.96.1-8.g.1.8, 56.96.1-8.g.1.9, 56.96.1-8.g.1.10, 56.96.1-8.g.1.11, 56.96.1-8.g.1.12, 88.96.1-8.g.1.1, 88.96.1-8.g.1.2, 88.96.1-8.g.1.3, 88.96.1-8.g.1.4, 88.96.1-8.g.1.5, 88.96.1-8.g.1.6, 88.96.1-8.g.1.7, 88.96.1-8.g.1.8, 88.96.1-8.g.1.9, 88.96.1-8.g.1.10, 88.96.1-8.g.1.11, 88.96.1-8.g.1.12, 104.96.1-8.g.1.1, 104.96.1-8.g.1.2, 104.96.1-8.g.1.3, 104.96.1-8.g.1.4, 104.96.1-8.g.1.5, 104.96.1-8.g.1.6, 104.96.1-8.g.1.7, 104.96.1-8.g.1.8, 104.96.1-8.g.1.9, 104.96.1-8.g.1.10, 104.96.1-8.g.1.11, 104.96.1-8.g.1.12, 120.96.1-8.g.1.1, 120.96.1-8.g.1.2, 120.96.1-8.g.1.3, 120.96.1-8.g.1.4, 120.96.1-8.g.1.5, 120.96.1-8.g.1.6, 120.96.1-8.g.1.7, 120.96.1-8.g.1.8, 120.96.1-8.g.1.9, 120.96.1-8.g.1.10, 120.96.1-8.g.1.11, 120.96.1-8.g.1.12, 136.96.1-8.g.1.1, 136.96.1-8.g.1.2, 136.96.1-8.g.1.3, 136.96.1-8.g.1.4, 136.96.1-8.g.1.5, 136.96.1-8.g.1.6, 136.96.1-8.g.1.7, 136.96.1-8.g.1.8, 136.96.1-8.g.1.9, 136.96.1-8.g.1.10, 136.96.1-8.g.1.11, 136.96.1-8.g.1.12, 152.96.1-8.g.1.1, 152.96.1-8.g.1.2, 152.96.1-8.g.1.3, 152.96.1-8.g.1.4, 152.96.1-8.g.1.5, 152.96.1-8.g.1.6, 152.96.1-8.g.1.7, 152.96.1-8.g.1.8, 152.96.1-8.g.1.9, 152.96.1-8.g.1.10, 152.96.1-8.g.1.11, 152.96.1-8.g.1.12, 168.96.1-8.g.1.1, 168.96.1-8.g.1.2, 168.96.1-8.g.1.3, 168.96.1-8.g.1.4, 168.96.1-8.g.1.5, 168.96.1-8.g.1.6, 168.96.1-8.g.1.7, 168.96.1-8.g.1.8, 168.96.1-8.g.1.9, 168.96.1-8.g.1.10, 168.96.1-8.g.1.11, 168.96.1-8.g.1.12, 184.96.1-8.g.1.1, 184.96.1-8.g.1.2, 184.96.1-8.g.1.3, 184.96.1-8.g.1.4, 184.96.1-8.g.1.5, 184.96.1-8.g.1.6, 184.96.1-8.g.1.7, 184.96.1-8.g.1.8, 184.96.1-8.g.1.9, 184.96.1-8.g.1.10, 184.96.1-8.g.1.11, 184.96.1-8.g.1.12, 232.96.1-8.g.1.1, 232.96.1-8.g.1.2, 232.96.1-8.g.1.3, 232.96.1-8.g.1.4, 232.96.1-8.g.1.5, 232.96.1-8.g.1.6, 232.96.1-8.g.1.7, 232.96.1-8.g.1.8, 232.96.1-8.g.1.9, 232.96.1-8.g.1.10, 232.96.1-8.g.1.11, 232.96.1-8.g.1.12, 248.96.1-8.g.1.1, 248.96.1-8.g.1.2, 248.96.1-8.g.1.3, 248.96.1-8.g.1.4, 248.96.1-8.g.1.5, 248.96.1-8.g.1.6, 248.96.1-8.g.1.7, 248.96.1-8.g.1.8, 248.96.1-8.g.1.9, 248.96.1-8.g.1.10, 248.96.1-8.g.1.11, 248.96.1-8.g.1.12, 264.96.1-8.g.1.1, 264.96.1-8.g.1.2, 264.96.1-8.g.1.3, 264.96.1-8.g.1.4, 264.96.1-8.g.1.5, 264.96.1-8.g.1.6, 264.96.1-8.g.1.7, 264.96.1-8.g.1.8, 264.96.1-8.g.1.9, 264.96.1-8.g.1.10, 264.96.1-8.g.1.11, 264.96.1-8.g.1.12, 280.96.1-8.g.1.1, 280.96.1-8.g.1.2, 280.96.1-8.g.1.3, 280.96.1-8.g.1.4, 280.96.1-8.g.1.5, 280.96.1-8.g.1.6, 280.96.1-8.g.1.7, 280.96.1-8.g.1.8, 280.96.1-8.g.1.9, 280.96.1-8.g.1.10, 280.96.1-8.g.1.11, 280.96.1-8.g.1.12, 296.96.1-8.g.1.1, 296.96.1-8.g.1.2, 296.96.1-8.g.1.3, 296.96.1-8.g.1.4, 296.96.1-8.g.1.5, 296.96.1-8.g.1.6, 296.96.1-8.g.1.7, 296.96.1-8.g.1.8, 296.96.1-8.g.1.9, 296.96.1-8.g.1.10, 296.96.1-8.g.1.11, 296.96.1-8.g.1.12, 312.96.1-8.g.1.1, 312.96.1-8.g.1.2, 312.96.1-8.g.1.3, 312.96.1-8.g.1.4, 312.96.1-8.g.1.5, 312.96.1-8.g.1.6, 312.96.1-8.g.1.7, 312.96.1-8.g.1.8, 312.96.1-8.g.1.9, 312.96.1-8.g.1.10, 312.96.1-8.g.1.11, 312.96.1-8.g.1.12, 328.96.1-8.g.1.1, 328.96.1-8.g.1.2, 328.96.1-8.g.1.3, 328.96.1-8.g.1.4, 328.96.1-8.g.1.5, 328.96.1-8.g.1.6, 328.96.1-8.g.1.7, 328.96.1-8.g.1.8, 328.96.1-8.g.1.9, 328.96.1-8.g.1.10, 328.96.1-8.g.1.11, 328.96.1-8.g.1.12
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $8$
Full 8-torsion field degree: $32$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{70x^{2}y^{12}z^{2}-4551x^{2}y^{8}z^{6}+27645x^{2}y^{4}z^{10}-4095x^{2}z^{14}-8xy^{14}z+1599xy^{10}z^{5}-25604xy^{6}z^{9}+20481xy^{2}z^{13}+y^{16}-308y^{12}z^{4}+10292y^{8}z^{8}-16386y^{4}z^{12}+z^{16}}{z^{2}y^{8}(x^{2}y^{4}-15x^{2}z^{4}+17xy^{2}z^{3}-6y^{4}z^{2}+z^{6})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{sp}}(4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.h.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.1.d.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
8.96.1.a.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.a.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.f.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.f.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.3.e.1 $8$ $2$ $2$ $3$ $0$ $1^{2}$
8.96.3.f.1 $8$ $2$ $2$ $3$ $0$ $2$
8.96.3.f.2 $8$ $2$ $2$ $3$ $0$ $2$
8.96.3.g.2 $8$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.1.e.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.e.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.p.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.p.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.o.2 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.96.3.p.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.p.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.q.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.144.9.cr.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.bh.2 $24$ $4$ $4$ $9$ $1$ $1^{8}$
40.96.1.e.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.e.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.p.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.p.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.3.t.2 $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.96.3.u.1 $40$ $2$ $2$ $3$ $0$ $2$
40.96.3.u.2 $40$ $2$ $2$ $3$ $0$ $2$
40.96.3.v.1 $40$ $2$ $2$ $3$ $2$ $1^{2}$
40.240.17.v.1 $40$ $5$ $5$ $17$ $5$ $1^{14}\cdot2$
40.288.17.bs.1 $40$ $6$ $6$ $17$ $2$ $1^{14}\cdot2$
40.480.33.et.1 $40$ $10$ $10$ $33$ $10$ $1^{28}\cdot2^{2}$
56.96.1.e.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.e.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.p.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.p.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.3.l.2 $56$ $2$ $2$ $3$ $2$ $1^{2}$
56.96.3.m.1 $56$ $2$ $2$ $3$ $0$ $2$
56.96.3.m.2 $56$ $2$ $2$ $3$ $0$ $2$
56.96.3.n.1 $56$ $2$ $2$ $3$ $2$ $1^{2}$
56.384.25.bh.1 $56$ $8$ $8$ $25$ $6$ $1^{20}\cdot2^{2}$
56.1008.73.cr.1 $56$ $21$ $21$ $73$ $23$ $1^{16}\cdot2^{26}\cdot4$
56.1344.97.cr.2 $56$ $28$ $28$ $97$ $29$ $1^{36}\cdot2^{28}\cdot4$
88.96.1.e.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.e.2 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.p.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.p.2 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.3.l.2 $88$ $2$ $2$ $3$ $?$ not computed
88.96.3.m.1 $88$ $2$ $2$ $3$ $?$ not computed
88.96.3.m.2 $88$ $2$ $2$ $3$ $?$ not computed
88.96.3.n.1 $88$ $2$ $2$ $3$ $?$ not computed
104.96.1.e.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.e.2 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.p.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.p.2 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.3.t.2 $104$ $2$ $2$ $3$ $?$ not computed
104.96.3.u.1 $104$ $2$ $2$ $3$ $?$ not computed
104.96.3.u.2 $104$ $2$ $2$ $3$ $?$ not computed
104.96.3.v.1 $104$ $2$ $2$ $3$ $?$ not computed
120.96.1.u.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.u.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bt.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bt.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.bp.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bq.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bq.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.br.1 $120$ $2$ $2$ $3$ $?$ not computed
136.96.1.e.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.e.2 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.p.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.p.2 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.3.t.2 $136$ $2$ $2$ $3$ $?$ not computed
136.96.3.u.1 $136$ $2$ $2$ $3$ $?$ not computed
136.96.3.u.2 $136$ $2$ $2$ $3$ $?$ not computed
136.96.3.v.1 $136$ $2$ $2$ $3$ $?$ not computed
152.96.1.e.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.e.2 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.p.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.p.2 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.3.l.2 $152$ $2$ $2$ $3$ $?$ not computed
152.96.3.m.1 $152$ $2$ $2$ $3$ $?$ not computed
152.96.3.m.2 $152$ $2$ $2$ $3$ $?$ not computed
152.96.3.n.1 $152$ $2$ $2$ $3$ $?$ not computed
168.96.1.u.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.u.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bt.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bt.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.bh.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bi.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bi.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bj.1 $168$ $2$ $2$ $3$ $?$ not computed
184.96.1.e.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.e.2 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.p.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.p.2 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.3.l.2 $184$ $2$ $2$ $3$ $?$ not computed
184.96.3.m.1 $184$ $2$ $2$ $3$ $?$ not computed
184.96.3.m.2 $184$ $2$ $2$ $3$ $?$ not computed
184.96.3.n.1 $184$ $2$ $2$ $3$ $?$ not computed
232.96.1.e.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.e.2 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.p.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.p.2 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.3.t.2 $232$ $2$ $2$ $3$ $?$ not computed
232.96.3.u.1 $232$ $2$ $2$ $3$ $?$ not computed
232.96.3.u.2 $232$ $2$ $2$ $3$ $?$ not computed
232.96.3.v.1 $232$ $2$ $2$ $3$ $?$ not computed
248.96.1.e.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.e.2 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.p.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.p.2 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.3.l.2 $248$ $2$ $2$ $3$ $?$ not computed
248.96.3.m.1 $248$ $2$ $2$ $3$ $?$ not computed
248.96.3.m.2 $248$ $2$ $2$ $3$ $?$ not computed
248.96.3.n.1 $248$ $2$ $2$ $3$ $?$ not computed
264.96.1.u.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.u.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.bt.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.bt.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.bh.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bi.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bi.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bj.1 $264$ $2$ $2$ $3$ $?$ not computed
280.96.1.u.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.u.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.bt.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.bt.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.3.bp.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.bq.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.bq.2 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.br.1 $280$ $2$ $2$ $3$ $?$ not computed
296.96.1.e.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.e.2 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.p.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.p.2 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.3.t.2 $296$ $2$ $2$ $3$ $?$ not computed
296.96.3.u.1 $296$ $2$ $2$ $3$ $?$ not computed
296.96.3.u.2 $296$ $2$ $2$ $3$ $?$ not computed
296.96.3.v.1 $296$ $2$ $2$ $3$ $?$ not computed
312.96.1.u.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.u.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.bt.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.bt.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.bp.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bq.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bq.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.br.1 $312$ $2$ $2$ $3$ $?$ not computed
328.96.1.e.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.e.2 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.p.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.p.2 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.3.t.2 $328$ $2$ $2$ $3$ $?$ not computed
328.96.3.u.1 $328$ $2$ $2$ $3$ $?$ not computed
328.96.3.u.2 $328$ $2$ $2$ $3$ $?$ not computed
328.96.3.v.1 $328$ $2$ $2$ $3$ $?$ not computed