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Properties

Label 8.48.1.bb.1
Level $8$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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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: $8$ (of which $4$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{4}\cdot2^{2}$
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: 8F1
Rouse and Zureick-Brown (RZB) label: X278
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.3

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}3&5\\0&5\end{bmatrix}$, $\begin{bmatrix}5&4\\0&5\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$, $\begin{bmatrix}7&5\\0&5\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.96.1-8.bb.1.1, 8.96.1-8.bb.1.2, 8.96.1-8.bb.1.3, 8.96.1-8.bb.1.4, 16.96.1-8.bb.1.1, 16.96.1-8.bb.1.2, 16.96.1-8.bb.1.3, 24.96.1-8.bb.1.1, 24.96.1-8.bb.1.2, 24.96.1-8.bb.1.3, 24.96.1-8.bb.1.4, 40.96.1-8.bb.1.1, 40.96.1-8.bb.1.2, 40.96.1-8.bb.1.3, 40.96.1-8.bb.1.4, 48.96.1-8.bb.1.1, 48.96.1-8.bb.1.2, 48.96.1-8.bb.1.3, 56.96.1-8.bb.1.1, 56.96.1-8.bb.1.2, 56.96.1-8.bb.1.3, 56.96.1-8.bb.1.4, 80.96.1-8.bb.1.1, 80.96.1-8.bb.1.2, 80.96.1-8.bb.1.3, 88.96.1-8.bb.1.1, 88.96.1-8.bb.1.2, 88.96.1-8.bb.1.3, 88.96.1-8.bb.1.4, 104.96.1-8.bb.1.1, 104.96.1-8.bb.1.2, 104.96.1-8.bb.1.3, 104.96.1-8.bb.1.4, 112.96.1-8.bb.1.1, 112.96.1-8.bb.1.2, 112.96.1-8.bb.1.3, 120.96.1-8.bb.1.1, 120.96.1-8.bb.1.2, 120.96.1-8.bb.1.3, 120.96.1-8.bb.1.4, 136.96.1-8.bb.1.1, 136.96.1-8.bb.1.2, 136.96.1-8.bb.1.3, 136.96.1-8.bb.1.4, 152.96.1-8.bb.1.1, 152.96.1-8.bb.1.2, 152.96.1-8.bb.1.3, 152.96.1-8.bb.1.4, 168.96.1-8.bb.1.1, 168.96.1-8.bb.1.2, 168.96.1-8.bb.1.3, 168.96.1-8.bb.1.4, 176.96.1-8.bb.1.1, 176.96.1-8.bb.1.2, 176.96.1-8.bb.1.3, 184.96.1-8.bb.1.1, 184.96.1-8.bb.1.2, 184.96.1-8.bb.1.3, 184.96.1-8.bb.1.4, 208.96.1-8.bb.1.1, 208.96.1-8.bb.1.2, 208.96.1-8.bb.1.3, 232.96.1-8.bb.1.1, 232.96.1-8.bb.1.2, 232.96.1-8.bb.1.3, 232.96.1-8.bb.1.4, 240.96.1-8.bb.1.1, 240.96.1-8.bb.1.2, 240.96.1-8.bb.1.3, 248.96.1-8.bb.1.1, 248.96.1-8.bb.1.2, 248.96.1-8.bb.1.3, 248.96.1-8.bb.1.4, 264.96.1-8.bb.1.1, 264.96.1-8.bb.1.2, 264.96.1-8.bb.1.3, 264.96.1-8.bb.1.4, 272.96.1-8.bb.1.1, 272.96.1-8.bb.1.2, 272.96.1-8.bb.1.3, 280.96.1-8.bb.1.1, 280.96.1-8.bb.1.2, 280.96.1-8.bb.1.3, 280.96.1-8.bb.1.4, 296.96.1-8.bb.1.1, 296.96.1-8.bb.1.2, 296.96.1-8.bb.1.3, 296.96.1-8.bb.1.4, 304.96.1-8.bb.1.1, 304.96.1-8.bb.1.2, 304.96.1-8.bb.1.3, 312.96.1-8.bb.1.1, 312.96.1-8.bb.1.2, 312.96.1-8.bb.1.3, 312.96.1-8.bb.1.4, 328.96.1-8.bb.1.1, 328.96.1-8.bb.1.2, 328.96.1-8.bb.1.3, 328.96.1-8.bb.1.4
Cyclic 8-isogeny field degree: $1$
Cyclic 8-torsion field degree: $4$
Full 8-torsion field degree: $32$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x $
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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
$(1:0:1)$, $(0:0:1)$, $(-1:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{9970x^{2}y^{12}z^{2}+97971x^{2}y^{8}z^{6}-18255x^{2}y^{4}z^{10}+4095x^{2}z^{14}-172xy^{14}z+161439xy^{10}z^{5}-112816xy^{6}z^{9}+20481xy^{2}z^{13}+y^{16}-200932y^{12}z^{4}-71428y^{8}z^{8}+16206y^{4}z^{12}+z^{16}}{zy^{4}(13x^{2}y^{8}z+501x^{2}y^{4}z^{5}+255x^{2}z^{9}+xy^{10}+268xy^{6}z^{4}+769xy^{2}z^{8}+70y^{8}z^{3}+522y^{4}z^{7}+z^{11})}$

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
4.24.0.c.1 $4$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.q.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.be.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.bf.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.1.m.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.w.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
8.96.1.l.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.l.2 $8$ $2$ $2$ $1$ $0$ dimension zero
16.96.3.ck.1 $16$ $2$ $2$ $3$ $0$ $2$
16.96.3.cn.1 $16$ $2$ $2$ $3$ $0$ $1^{2}$
16.96.3.cu.1 $16$ $2$ $2$ $3$ $0$ $2$
24.96.1.cs.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cs.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.uu.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.hw.1 $24$ $4$ $4$ $9$ $0$ $1^{8}$
40.96.1.cj.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.cj.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.fe.1 $40$ $5$ $5$ $17$ $3$ $1^{14}\cdot2$
40.288.17.mt.1 $40$ $6$ $6$ $17$ $1$ $1^{14}\cdot2$
40.480.33.xg.1 $40$ $10$ $10$ $33$ $7$ $1^{28}\cdot2^{2}$
48.96.3.fl.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.fp.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.ge.1 $48$ $2$ $2$ $3$ $0$ $2$
56.96.1.cj.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.cj.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.hw.1 $56$ $8$ $8$ $25$ $3$ $1^{20}\cdot2^{2}$
56.1008.73.uu.1 $56$ $21$ $21$ $73$ $19$ $1^{16}\cdot2^{26}\cdot4$
56.1344.97.um.1 $56$ $28$ $28$ $97$ $22$ $1^{36}\cdot2^{28}\cdot4$
80.96.3.gn.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.gr.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.hg.1 $80$ $2$ $2$ $3$ $?$ not computed
88.96.1.cj.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.cj.2 $88$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.cj.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.cj.2 $104$ $2$ $2$ $1$ $?$ dimension zero
112.96.3.fl.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.fp.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.ge.1 $112$ $2$ $2$ $3$ $?$ not computed
120.96.1.qi.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.qi.2 $120$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.cj.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.cj.2 $136$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.cj.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.cj.2 $152$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qi.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qi.2 $168$ $2$ $2$ $1$ $?$ dimension zero
176.96.3.fl.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.fp.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.ge.1 $176$ $2$ $2$ $3$ $?$ not computed
184.96.1.cj.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.cj.2 $184$ $2$ $2$ $1$ $?$ dimension zero
208.96.3.gn.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.gr.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.hg.1 $208$ $2$ $2$ $3$ $?$ not computed
232.96.1.cj.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.cj.2 $232$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.qv.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.rd.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.sq.1 $240$ $2$ $2$ $3$ $?$ not computed
248.96.1.cj.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.cj.2 $248$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qi.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qi.2 $264$ $2$ $2$ $1$ $?$ dimension zero
272.96.3.gn.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.gr.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.hg.1 $272$ $2$ $2$ $3$ $?$ not computed
280.96.1.pn.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.pn.2 $280$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.cj.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.cj.2 $296$ $2$ $2$ $1$ $?$ dimension zero
304.96.3.fl.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.fp.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.ge.1 $304$ $2$ $2$ $3$ $?$ not computed
312.96.1.qi.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.qi.2 $312$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.cj.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.cj.2 $328$ $2$ $2$ $1$ $?$ dimension zero