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Properties

Label 8.48.1.i.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: $32$
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 $2$ 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: 8G1
Rouse and Zureick-Brown (RZB) label: X272
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.33

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}3&2\\0&5\end{bmatrix}$, $\begin{bmatrix}5&2\\0&3\end{bmatrix}$, $\begin{bmatrix}7&0\\4&1\end{bmatrix}$, $\begin{bmatrix}7&0\\4&5\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.96.1-8.i.1.1, 8.96.1-8.i.1.2, 8.96.1-8.i.1.3, 8.96.1-8.i.1.4, 8.96.1-8.i.1.5, 8.96.1-8.i.1.6, 8.96.1-8.i.1.7, 8.96.1-8.i.1.8, 16.96.1-8.i.1.1, 16.96.1-8.i.1.2, 16.96.1-8.i.1.3, 16.96.1-8.i.1.4, 24.96.1-8.i.1.1, 24.96.1-8.i.1.2, 24.96.1-8.i.1.3, 24.96.1-8.i.1.4, 24.96.1-8.i.1.5, 24.96.1-8.i.1.6, 24.96.1-8.i.1.7, 24.96.1-8.i.1.8, 40.96.1-8.i.1.1, 40.96.1-8.i.1.2, 40.96.1-8.i.1.3, 40.96.1-8.i.1.4, 40.96.1-8.i.1.5, 40.96.1-8.i.1.6, 40.96.1-8.i.1.7, 40.96.1-8.i.1.8, 48.96.1-8.i.1.1, 48.96.1-8.i.1.2, 48.96.1-8.i.1.3, 48.96.1-8.i.1.4, 56.96.1-8.i.1.1, 56.96.1-8.i.1.2, 56.96.1-8.i.1.3, 56.96.1-8.i.1.4, 56.96.1-8.i.1.5, 56.96.1-8.i.1.6, 56.96.1-8.i.1.7, 56.96.1-8.i.1.8, 80.96.1-8.i.1.1, 80.96.1-8.i.1.2, 80.96.1-8.i.1.3, 80.96.1-8.i.1.4, 88.96.1-8.i.1.1, 88.96.1-8.i.1.2, 88.96.1-8.i.1.3, 88.96.1-8.i.1.4, 88.96.1-8.i.1.5, 88.96.1-8.i.1.6, 88.96.1-8.i.1.7, 88.96.1-8.i.1.8, 104.96.1-8.i.1.1, 104.96.1-8.i.1.2, 104.96.1-8.i.1.3, 104.96.1-8.i.1.4, 104.96.1-8.i.1.5, 104.96.1-8.i.1.6, 104.96.1-8.i.1.7, 104.96.1-8.i.1.8, 112.96.1-8.i.1.1, 112.96.1-8.i.1.2, 112.96.1-8.i.1.3, 112.96.1-8.i.1.4, 120.96.1-8.i.1.1, 120.96.1-8.i.1.2, 120.96.1-8.i.1.3, 120.96.1-8.i.1.4, 120.96.1-8.i.1.5, 120.96.1-8.i.1.6, 120.96.1-8.i.1.7, 120.96.1-8.i.1.8, 136.96.1-8.i.1.1, 136.96.1-8.i.1.2, 136.96.1-8.i.1.3, 136.96.1-8.i.1.4, 136.96.1-8.i.1.5, 136.96.1-8.i.1.6, 136.96.1-8.i.1.7, 136.96.1-8.i.1.8, 152.96.1-8.i.1.1, 152.96.1-8.i.1.2, 152.96.1-8.i.1.3, 152.96.1-8.i.1.4, 152.96.1-8.i.1.5, 152.96.1-8.i.1.6, 152.96.1-8.i.1.7, 152.96.1-8.i.1.8, 168.96.1-8.i.1.1, 168.96.1-8.i.1.2, 168.96.1-8.i.1.3, 168.96.1-8.i.1.4, 168.96.1-8.i.1.5, 168.96.1-8.i.1.6, 168.96.1-8.i.1.7, 168.96.1-8.i.1.8, 176.96.1-8.i.1.1, 176.96.1-8.i.1.2, 176.96.1-8.i.1.3, 176.96.1-8.i.1.4, 184.96.1-8.i.1.1, 184.96.1-8.i.1.2, 184.96.1-8.i.1.3, 184.96.1-8.i.1.4, 184.96.1-8.i.1.5, 184.96.1-8.i.1.6, 184.96.1-8.i.1.7, 184.96.1-8.i.1.8, 208.96.1-8.i.1.1, 208.96.1-8.i.1.2, 208.96.1-8.i.1.3, 208.96.1-8.i.1.4, 232.96.1-8.i.1.1, 232.96.1-8.i.1.2, 232.96.1-8.i.1.3, 232.96.1-8.i.1.4, 232.96.1-8.i.1.5, 232.96.1-8.i.1.6, 232.96.1-8.i.1.7, 232.96.1-8.i.1.8, 240.96.1-8.i.1.1, 240.96.1-8.i.1.2, 240.96.1-8.i.1.3, 240.96.1-8.i.1.4, 248.96.1-8.i.1.1, 248.96.1-8.i.1.2, 248.96.1-8.i.1.3, 248.96.1-8.i.1.4, 248.96.1-8.i.1.5, 248.96.1-8.i.1.6, 248.96.1-8.i.1.7, 248.96.1-8.i.1.8, 264.96.1-8.i.1.1, 264.96.1-8.i.1.2, 264.96.1-8.i.1.3, 264.96.1-8.i.1.4, 264.96.1-8.i.1.5, 264.96.1-8.i.1.6, 264.96.1-8.i.1.7, 264.96.1-8.i.1.8, 272.96.1-8.i.1.1, 272.96.1-8.i.1.2, 272.96.1-8.i.1.3, 272.96.1-8.i.1.4, 280.96.1-8.i.1.1, 280.96.1-8.i.1.2, 280.96.1-8.i.1.3, 280.96.1-8.i.1.4, 280.96.1-8.i.1.5, 280.96.1-8.i.1.6, 280.96.1-8.i.1.7, 280.96.1-8.i.1.8, 296.96.1-8.i.1.1, 296.96.1-8.i.1.2, 296.96.1-8.i.1.3, 296.96.1-8.i.1.4, 296.96.1-8.i.1.5, 296.96.1-8.i.1.6, 296.96.1-8.i.1.7, 296.96.1-8.i.1.8, 304.96.1-8.i.1.1, 304.96.1-8.i.1.2, 304.96.1-8.i.1.3, 304.96.1-8.i.1.4, 312.96.1-8.i.1.1, 312.96.1-8.i.1.2, 312.96.1-8.i.1.3, 312.96.1-8.i.1.4, 312.96.1-8.i.1.5, 312.96.1-8.i.1.6, 312.96.1-8.i.1.7, 312.96.1-8.i.1.8, 328.96.1-8.i.1.1, 328.96.1-8.i.1.2, 328.96.1-8.i.1.3, 328.96.1-8.i.1.4, 328.96.1-8.i.1.5, 328.96.1-8.i.1.6, 328.96.1-8.i.1.7, 328.96.1-8.i.1.8
Cyclic 8-isogeny field degree: $2$
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} - 11x - 14 $
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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
$(-2:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{24x^{2}y^{14}+22306x^{2}y^{12}z^{2}+5483544x^{2}y^{10}z^{4}+706612089x^{2}y^{8}z^{6}+54140076096x^{2}y^{6}z^{8}+2518894510089x^{2}y^{4}z^{10}+66838641770484x^{2}y^{2}z^{12}+782005818097665x^{2}z^{14}+316xy^{14}z+163752xy^{12}z^{3}+33671679xy^{10}z^{5}+3831229002xy^{8}z^{7}+266884611976xy^{6}z^{9}+11420030730216xy^{4}z^{11}+279605172764697xy^{2}z^{13}+2993852285714430xz^{15}+y^{16}+2832y^{14}z^{2}+896388y^{12}z^{4}+139544184y^{10}z^{6}+12584987372y^{8}z^{8}+700717202976y^{6}z^{10}+23533979697114y^{4}z^{12}+426026765123664y^{2}z^{14}+2859681299038201z^{16}}{z^{2}y^{4}(x^{2}y^{8}+2836x^{2}y^{6}z^{2}+629505x^{2}y^{4}z^{4}+37916672x^{2}y^{2}z^{6}+661766144x^{2}z^{8}+24xy^{8}z+21721xy^{6}z^{3}+3347966xy^{4}z^{5}+165232640xy^{2}z^{7}+2533523456xz^{9}+306y^{8}z^{2}+115568y^{6}z^{4}+10092537y^{4}z^{6}+292339712y^{2}z^{8}+2419982336z^{10})}$

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
8.24.0.d.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.e.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.1.c.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.b.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.c.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.g.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.h.1 $8$ $2$ $2$ $1$ $0$ dimension zero
16.96.5.m.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.96.5.n.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.96.5.p.1 $16$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
16.96.5.q.1 $16$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.96.1.i.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.j.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.t.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.u.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.df.1 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
24.192.9.bs.2 $24$ $4$ $4$ $9$ $0$ $1^{4}\cdot2^{2}$
40.96.1.i.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.j.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.t.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.u.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.bb.1 $40$ $5$ $5$ $17$ $3$ $1^{6}\cdot2^{5}$
40.288.17.cc.1 $40$ $6$ $6$ $17$ $1$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.fh.1 $40$ $10$ $10$ $33$ $5$ $1^{12}\cdot2^{6}\cdot4^{2}$
48.96.5.bj.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.bk.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.bp.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.bq.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
56.96.1.i.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.j.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.t.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.u.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.bs.2 $56$ $8$ $8$ $25$ $1$ $1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.df.1 $56$ $21$ $21$ $73$ $9$ $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.df.1 $56$ $28$ $28$ $97$ $10$ $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
80.96.5.bj.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.bk.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.bp.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.bq.2 $80$ $2$ $2$ $5$ $?$ not computed
88.96.1.i.2 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.j.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.t.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.u.1 $88$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.i.2 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.j.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.t.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.u.2 $104$ $2$ $2$ $1$ $?$ dimension zero
112.96.5.bj.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.bk.2 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.bp.1 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.bq.2 $112$ $2$ $2$ $5$ $?$ not computed
120.96.1.be.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bf.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.cd.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ce.1 $120$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.i.2 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.j.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.t.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.u.2 $136$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.i.2 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.j.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.t.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.u.2 $152$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.be.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bf.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.cd.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ce.2 $168$ $2$ $2$ $1$ $?$ dimension zero
176.96.5.bj.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.bk.2 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.bp.1 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.bq.2 $176$ $2$ $2$ $5$ $?$ not computed
184.96.1.i.2 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.j.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.t.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.u.1 $184$ $2$ $2$ $1$ $?$ dimension zero
208.96.5.bj.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.bk.2 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.bp.1 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.bq.2 $208$ $2$ $2$ $5$ $?$ not computed
232.96.1.i.2 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.j.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.t.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.u.2 $232$ $2$ $2$ $1$ $?$ dimension zero
240.96.5.eb.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ec.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.eo.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ep.2 $240$ $2$ $2$ $5$ $?$ not computed
248.96.1.i.2 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.j.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.t.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.u.1 $248$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.be.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.bf.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.cd.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ce.1 $264$ $2$ $2$ $1$ $?$ dimension zero
272.96.5.bj.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.bk.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.bp.1 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.bq.1 $272$ $2$ $2$ $5$ $?$ not computed
280.96.1.be.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.bf.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.cd.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ce.1 $280$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.i.2 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.j.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.t.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.u.2 $296$ $2$ $2$ $1$ $?$ dimension zero
304.96.5.bj.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.bk.2 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.bp.1 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.bq.2 $304$ $2$ $2$ $5$ $?$ not computed
312.96.1.be.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.bf.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.cd.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ce.2 $312$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.i.2 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.j.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.t.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.u.2 $328$ $2$ $2$ $1$ $?$ dimension zero