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Properties

Label 8.48.1.k.2
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: 8G1
Rouse and Zureick-Brown (RZB) label: X245
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.51

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&0\\4&7\end{bmatrix}$, $\begin{bmatrix}5&0\\4&7\end{bmatrix}$, $\begin{bmatrix}5&6\\4&1\end{bmatrix}$, $\begin{bmatrix}7&0\\4&1\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.96.1-8.k.2.1, 8.96.1-8.k.2.2, 8.96.1-8.k.2.3, 8.96.1-8.k.2.4, 8.96.1-8.k.2.5, 8.96.1-8.k.2.6, 16.96.1-8.k.2.1, 16.96.1-8.k.2.2, 16.96.1-8.k.2.3, 16.96.1-8.k.2.4, 16.96.1-8.k.2.5, 16.96.1-8.k.2.6, 16.96.1-8.k.2.7, 16.96.1-8.k.2.8, 24.96.1-8.k.2.1, 24.96.1-8.k.2.2, 24.96.1-8.k.2.3, 24.96.1-8.k.2.4, 24.96.1-8.k.2.5, 24.96.1-8.k.2.6, 40.96.1-8.k.2.1, 40.96.1-8.k.2.2, 40.96.1-8.k.2.3, 40.96.1-8.k.2.4, 40.96.1-8.k.2.5, 40.96.1-8.k.2.6, 48.96.1-8.k.2.1, 48.96.1-8.k.2.2, 48.96.1-8.k.2.3, 48.96.1-8.k.2.4, 48.96.1-8.k.2.5, 48.96.1-8.k.2.6, 48.96.1-8.k.2.7, 48.96.1-8.k.2.8, 56.96.1-8.k.2.1, 56.96.1-8.k.2.2, 56.96.1-8.k.2.3, 56.96.1-8.k.2.4, 56.96.1-8.k.2.5, 56.96.1-8.k.2.6, 80.96.1-8.k.2.1, 80.96.1-8.k.2.2, 80.96.1-8.k.2.3, 80.96.1-8.k.2.4, 80.96.1-8.k.2.5, 80.96.1-8.k.2.6, 80.96.1-8.k.2.7, 80.96.1-8.k.2.8, 88.96.1-8.k.2.1, 88.96.1-8.k.2.2, 88.96.1-8.k.2.3, 88.96.1-8.k.2.4, 88.96.1-8.k.2.5, 88.96.1-8.k.2.6, 104.96.1-8.k.2.1, 104.96.1-8.k.2.2, 104.96.1-8.k.2.3, 104.96.1-8.k.2.4, 104.96.1-8.k.2.5, 104.96.1-8.k.2.6, 112.96.1-8.k.2.1, 112.96.1-8.k.2.2, 112.96.1-8.k.2.3, 112.96.1-8.k.2.4, 112.96.1-8.k.2.5, 112.96.1-8.k.2.6, 112.96.1-8.k.2.7, 112.96.1-8.k.2.8, 120.96.1-8.k.2.1, 120.96.1-8.k.2.2, 120.96.1-8.k.2.3, 120.96.1-8.k.2.4, 120.96.1-8.k.2.5, 120.96.1-8.k.2.6, 136.96.1-8.k.2.1, 136.96.1-8.k.2.2, 136.96.1-8.k.2.3, 136.96.1-8.k.2.4, 136.96.1-8.k.2.5, 136.96.1-8.k.2.6, 152.96.1-8.k.2.1, 152.96.1-8.k.2.2, 152.96.1-8.k.2.3, 152.96.1-8.k.2.4, 152.96.1-8.k.2.5, 152.96.1-8.k.2.6, 168.96.1-8.k.2.1, 168.96.1-8.k.2.2, 168.96.1-8.k.2.3, 168.96.1-8.k.2.4, 168.96.1-8.k.2.5, 168.96.1-8.k.2.6, 176.96.1-8.k.2.1, 176.96.1-8.k.2.2, 176.96.1-8.k.2.3, 176.96.1-8.k.2.4, 176.96.1-8.k.2.5, 176.96.1-8.k.2.6, 176.96.1-8.k.2.7, 176.96.1-8.k.2.8, 184.96.1-8.k.2.1, 184.96.1-8.k.2.2, 184.96.1-8.k.2.3, 184.96.1-8.k.2.4, 184.96.1-8.k.2.5, 184.96.1-8.k.2.6, 208.96.1-8.k.2.1, 208.96.1-8.k.2.2, 208.96.1-8.k.2.3, 208.96.1-8.k.2.4, 208.96.1-8.k.2.5, 208.96.1-8.k.2.6, 208.96.1-8.k.2.7, 208.96.1-8.k.2.8, 232.96.1-8.k.2.1, 232.96.1-8.k.2.2, 232.96.1-8.k.2.3, 232.96.1-8.k.2.4, 232.96.1-8.k.2.5, 232.96.1-8.k.2.6, 240.96.1-8.k.2.1, 240.96.1-8.k.2.2, 240.96.1-8.k.2.3, 240.96.1-8.k.2.4, 240.96.1-8.k.2.5, 240.96.1-8.k.2.6, 240.96.1-8.k.2.7, 240.96.1-8.k.2.8, 248.96.1-8.k.2.1, 248.96.1-8.k.2.2, 248.96.1-8.k.2.3, 248.96.1-8.k.2.4, 248.96.1-8.k.2.5, 248.96.1-8.k.2.6, 264.96.1-8.k.2.1, 264.96.1-8.k.2.2, 264.96.1-8.k.2.3, 264.96.1-8.k.2.4, 264.96.1-8.k.2.5, 264.96.1-8.k.2.6, 272.96.1-8.k.2.1, 272.96.1-8.k.2.2, 272.96.1-8.k.2.3, 272.96.1-8.k.2.4, 272.96.1-8.k.2.5, 272.96.1-8.k.2.6, 272.96.1-8.k.2.7, 272.96.1-8.k.2.8, 280.96.1-8.k.2.1, 280.96.1-8.k.2.2, 280.96.1-8.k.2.3, 280.96.1-8.k.2.4, 280.96.1-8.k.2.5, 280.96.1-8.k.2.6, 296.96.1-8.k.2.1, 296.96.1-8.k.2.2, 296.96.1-8.k.2.3, 296.96.1-8.k.2.4, 296.96.1-8.k.2.5, 296.96.1-8.k.2.6, 304.96.1-8.k.2.1, 304.96.1-8.k.2.2, 304.96.1-8.k.2.3, 304.96.1-8.k.2.4, 304.96.1-8.k.2.5, 304.96.1-8.k.2.6, 304.96.1-8.k.2.7, 304.96.1-8.k.2.8, 312.96.1-8.k.2.1, 312.96.1-8.k.2.2, 312.96.1-8.k.2.3, 312.96.1-8.k.2.4, 312.96.1-8.k.2.5, 312.96.1-8.k.2.6, 328.96.1-8.k.2.1, 328.96.1-8.k.2.2, 328.96.1-8.k.2.3, 328.96.1-8.k.2.4, 328.96.1-8.k.2.5, 328.96.1-8.k.2.6
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

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 11x + 14 $
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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
$(2:0:1)$, $(0:1:0)$, $(1:-2:1)$, $(1:2:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{24x^{2}y^{14}-424786x^{2}y^{12}z^{2}+1011536664x^{2}y^{10}z^{4}-588578534409x^{2}y^{8}z^{6}+109936935701376x^{2}y^{6}z^{8}-8365764142695129x^{2}y^{4}z^{10}+273771076691951604x^{2}y^{2}z^{12}-3203095830928031745x^{2}z^{14}-1036xy^{14}z+6584712xy^{12}z^{3}-10157502399xy^{10}z^{5}+4099344479562xy^{8}z^{7}-605868487863256xy^{6}z^{9}+39304781176502856xy^{4}z^{11}-1145262787644096537xy^{2}z^{13}+12262818962286313470xz^{15}-y^{16}+20112y^{14}z^{2}-86989668y^{12}z^{4}+78616007544y^{10}z^{6}-20141247406412y^{8}z^{8}+2007992773878816y^{6}z^{10}-89258362532785194y^{4}z^{12}+1745005629946200144y^{2}z^{14}-11713254600860499961z^{16}}{zy^{4}(287x^{2}y^{8}z-66800x^{2}y^{6}z^{3}+2293821x^{2}y^{4}z^{5}-4x^{2}y^{2}z^{7}+x^{2}z^{9}+xy^{10}-2390xy^{8}z^{2}+325308xy^{6}z^{4}-8781712xy^{4}z^{6}-7xy^{2}z^{8}+2xz^{10}-24y^{10}z+13046y^{8}z^{3}-776976y^{6}z^{5}+8388142y^{4}z^{7}+32y^{2}z^{9}-7z^{11})}$

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.e.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.bl.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.g.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.h.1 $8$ $2$ $2$ $1$ $0$ dimension zero
16.96.3.m.2 $16$ $2$ $2$ $3$ $0$ $2$
16.96.3.n.2 $16$ $2$ $2$ $3$ $0$ $2$
16.96.3.bc.2 $16$ $2$ $2$ $3$ $0$ $2$
16.96.3.bd.2 $16$ $2$ $2$ $3$ $0$ $2$
16.96.5.t.2 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.96.5.u.2 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.96.5.v.2 $16$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
16.96.5.w.2 $16$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.96.1.y.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.z.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.dn.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
24.192.9.by.2 $24$ $4$ $4$ $9$ $0$ $1^{4}\cdot2^{2}$
40.96.1.y.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.z.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.bf.2 $40$ $5$ $5$ $17$ $1$ $1^{6}\cdot2^{5}$
40.288.17.ci.1 $40$ $6$ $6$ $17$ $1$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.fp.2 $40$ $10$ $10$ $33$ $3$ $1^{12}\cdot2^{6}\cdot4^{2}$
48.96.3.be.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bf.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.by.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bz.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.5.ch.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.ci.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.cj.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.ck.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
56.96.1.y.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.z.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.by.1 $56$ $8$ $8$ $25$ $1$ $1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.dn.2 $56$ $21$ $21$ $73$ $9$ $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.dn.1 $56$ $28$ $28$ $97$ $10$ $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
80.96.3.bs.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.bt.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.co.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.cp.2 $80$ $2$ $2$ $3$ $?$ not computed
80.96.5.ch.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ci.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.cj.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ck.2 $80$ $2$ $2$ $5$ $?$ not computed
88.96.1.y.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.z.2 $88$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.y.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.z.1 $104$ $2$ $2$ $1$ $?$ dimension zero
112.96.3.be.2 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.bf.2 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.by.2 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.bz.2 $112$ $2$ $2$ $3$ $?$ not computed
112.96.5.ch.2 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.ci.2 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.cj.2 $112$ $2$ $2$ $5$ $?$ not computed
112.96.5.ck.2 $112$ $2$ $2$ $5$ $?$ not computed
120.96.1.da.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.db.1 $120$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.y.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.z.1 $136$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.y.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.z.2 $152$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.da.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.db.1 $168$ $2$ $2$ $1$ $?$ dimension zero
176.96.3.be.2 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.bf.2 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.by.2 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.bz.2 $176$ $2$ $2$ $3$ $?$ not computed
176.96.5.ch.2 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.ci.2 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.cj.2 $176$ $2$ $2$ $5$ $?$ not computed
176.96.5.ck.2 $176$ $2$ $2$ $5$ $?$ not computed
184.96.1.y.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.z.2 $184$ $2$ $2$ $1$ $?$ dimension zero
208.96.3.bq.2 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.br.2 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.cq.2 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.cr.2 $208$ $2$ $2$ $3$ $?$ not computed
208.96.5.ch.2 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.ci.2 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.cj.2 $208$ $2$ $2$ $5$ $?$ not computed
208.96.5.ck.2 $208$ $2$ $2$ $5$ $?$ not computed
232.96.1.y.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.z.1 $232$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.do.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.dp.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ek.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.el.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.5.gx.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.gy.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.gz.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ha.2 $240$ $2$ $2$ $5$ $?$ not computed
248.96.1.y.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.z.2 $248$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.da.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.db.2 $264$ $2$ $2$ $1$ $?$ dimension zero
272.96.3.bo.2 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.bp.2 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.cs.2 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.ct.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.5.ch.2 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.ci.2 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.cj.2 $272$ $2$ $2$ $5$ $?$ not computed
272.96.5.ck.2 $272$ $2$ $2$ $5$ $?$ not computed
280.96.1.da.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.db.1 $280$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.y.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.z.1 $296$ $2$ $2$ $1$ $?$ dimension zero
304.96.3.be.2 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.bf.2 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.by.2 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.bz.2 $304$ $2$ $2$ $3$ $?$ not computed
304.96.5.ch.2 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.ci.2 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.cj.2 $304$ $2$ $2$ $5$ $?$ not computed
304.96.5.ck.2 $304$ $2$ $2$ $5$ $?$ not computed
312.96.1.da.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.db.1 $312$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.y.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.z.1 $328$ $2$ $2$ $1$ $?$ dimension zero