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Properties

Label 8.24.0.bo.1
Level $8$
Index $24$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $8$ $\SL_2$-level: $8$
Index: $24$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $4^{2}\cdot8^{2}$ Cusp orbits $2^{2}$
Elliptic points: $4$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $0$
Rational CM points: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 8H0
Sutherland and Zywina (SZ) label: 8H0-8j
Rouse and Zureick-Brown (RZB) label: X97
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.24.0.134

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&1\\0&7\end{bmatrix}$, $\begin{bmatrix}1&5\\6&7\end{bmatrix}$, $\begin{bmatrix}1&6\\2&7\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^3.C_2^3$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 8-isogeny field degree: $4$
Cyclic 8-torsion field degree: $16$
Full 8-torsion field degree: $64$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 24 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^6\,\frac{(2x+y)^{24}(2x^{2}+4xy+3y^{2})^{3}(6x^{2}+4xy+y^{2})^{3}(4x^{4}+32x^{3}y+44x^{2}y^{2}+16xy^{3}+y^{4})^{3}}{(2x+y)^{24}(2x^{2}-y^{2})^{8}(2x^{2}+4xy+y^{2})^{4}}$

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
8.12.0.s.1 $8$ $2$ $2$ $0$ $0$
8.12.0.v.1 $8$ $2$ $2$ $0$ $0$
8.12.0.y.1 $8$ $2$ $2$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
8.48.1.c.1 $8$ $2$ $2$ $1$
8.48.1.t.1 $8$ $2$ $2$ $1$
8.48.1.bn.1 $8$ $2$ $2$ $1$
8.48.1.bp.1 $8$ $2$ $2$ $1$
16.48.1.y.1 $16$ $2$ $2$ $1$
16.48.1.ba.1 $16$ $2$ $2$ $1$
16.48.1.bq.1 $16$ $2$ $2$ $1$
16.48.1.br.1 $16$ $2$ $2$ $1$
16.48.1.bs.1 $16$ $2$ $2$ $1$
16.48.1.bt.1 $16$ $2$ $2$ $1$
16.48.1.ca.1 $16$ $2$ $2$ $1$
16.48.1.cc.1 $16$ $2$ $2$ $1$
16.48.2.bw.1 $16$ $2$ $2$ $2$
16.48.2.bx.1 $16$ $2$ $2$ $2$
24.48.1.la.1 $24$ $2$ $2$ $1$
24.48.1.le.1 $24$ $2$ $2$ $1$
24.48.1.lq.1 $24$ $2$ $2$ $1$
24.48.1.lu.1 $24$ $2$ $2$ $1$
24.72.2.jo.1 $24$ $3$ $3$ $2$
24.96.5.io.1 $24$ $4$ $4$ $5$
40.48.1.iy.1 $40$ $2$ $2$ $1$
40.48.1.jc.1 $40$ $2$ $2$ $1$
40.48.1.jo.1 $40$ $2$ $2$ $1$
40.48.1.js.1 $40$ $2$ $2$ $1$
40.120.8.gu.1 $40$ $5$ $5$ $8$
40.144.7.ri.1 $40$ $6$ $6$ $7$
40.240.15.bby.1 $40$ $10$ $10$ $15$
48.48.1.fq.1 $48$ $2$ $2$ $1$
48.48.1.fs.1 $48$ $2$ $2$ $1$
48.48.1.fy.1 $48$ $2$ $2$ $1$
48.48.1.fz.1 $48$ $2$ $2$ $1$
48.48.1.ga.1 $48$ $2$ $2$ $1$
48.48.1.gb.1 $48$ $2$ $2$ $1$
48.48.1.gi.1 $48$ $2$ $2$ $1$
48.48.1.gk.1 $48$ $2$ $2$ $1$
48.48.2.ey.1 $48$ $2$ $2$ $2$
48.48.2.ez.1 $48$ $2$ $2$ $2$
56.48.1.iw.1 $56$ $2$ $2$ $1$
56.48.1.ja.1 $56$ $2$ $2$ $1$
56.48.1.jm.1 $56$ $2$ $2$ $1$
56.48.1.jq.1 $56$ $2$ $2$ $1$
56.192.13.hq.1 $56$ $8$ $8$ $13$
56.504.32.fa.1 $56$ $21$ $21$ $32$
56.672.45.bbe.1 $56$ $28$ $28$ $45$
80.48.1.fs.1 $80$ $2$ $2$ $1$
80.48.1.fu.1 $80$ $2$ $2$ $1$
80.48.1.ga.1 $80$ $2$ $2$ $1$
80.48.1.gb.1 $80$ $2$ $2$ $1$
80.48.1.gc.1 $80$ $2$ $2$ $1$
80.48.1.gd.1 $80$ $2$ $2$ $1$
80.48.1.gk.1 $80$ $2$ $2$ $1$
80.48.1.gm.1 $80$ $2$ $2$ $1$
80.48.2.fi.1 $80$ $2$ $2$ $2$
80.48.2.fj.1 $80$ $2$ $2$ $2$
88.48.1.iw.1 $88$ $2$ $2$ $1$
88.48.1.ja.1 $88$ $2$ $2$ $1$
88.48.1.jm.1 $88$ $2$ $2$ $1$
88.48.1.jq.1 $88$ $2$ $2$ $1$
88.288.21.hq.1 $88$ $12$ $12$ $21$
104.48.1.iy.1 $104$ $2$ $2$ $1$
104.48.1.jc.1 $104$ $2$ $2$ $1$
104.48.1.jo.1 $104$ $2$ $2$ $1$
104.48.1.js.1 $104$ $2$ $2$ $1$
104.336.23.sm.1 $104$ $14$ $14$ $23$
112.48.1.fq.1 $112$ $2$ $2$ $1$
112.48.1.fs.1 $112$ $2$ $2$ $1$
112.48.1.fy.1 $112$ $2$ $2$ $1$
112.48.1.fz.1 $112$ $2$ $2$ $1$
112.48.1.ga.1 $112$ $2$ $2$ $1$
112.48.1.gb.1 $112$ $2$ $2$ $1$
112.48.1.gi.1 $112$ $2$ $2$ $1$
112.48.1.gk.1 $112$ $2$ $2$ $1$
112.48.2.ey.1 $112$ $2$ $2$ $2$
112.48.2.ez.1 $112$ $2$ $2$ $2$
120.48.1.cqu.1 $120$ $2$ $2$ $1$
120.48.1.cqy.1 $120$ $2$ $2$ $1$
120.48.1.csa.1 $120$ $2$ $2$ $1$
120.48.1.cse.1 $120$ $2$ $2$ $1$
136.48.1.iy.1 $136$ $2$ $2$ $1$
136.48.1.jc.1 $136$ $2$ $2$ $1$
136.48.1.jo.1 $136$ $2$ $2$ $1$
136.48.1.js.1 $136$ $2$ $2$ $1$
152.48.1.iw.1 $152$ $2$ $2$ $1$
152.48.1.ja.1 $152$ $2$ $2$ $1$
152.48.1.jm.1 $152$ $2$ $2$ $1$
152.48.1.jq.1 $152$ $2$ $2$ $1$
168.48.1.cqo.1 $168$ $2$ $2$ $1$
168.48.1.cqs.1 $168$ $2$ $2$ $1$
168.48.1.cru.1 $168$ $2$ $2$ $1$
168.48.1.cry.1 $168$ $2$ $2$ $1$
176.48.1.fq.1 $176$ $2$ $2$ $1$
176.48.1.fs.1 $176$ $2$ $2$ $1$
176.48.1.fy.1 $176$ $2$ $2$ $1$
176.48.1.fz.1 $176$ $2$ $2$ $1$
176.48.1.ga.1 $176$ $2$ $2$ $1$
176.48.1.gb.1 $176$ $2$ $2$ $1$
176.48.1.gi.1 $176$ $2$ $2$ $1$
176.48.1.gk.1 $176$ $2$ $2$ $1$
176.48.2.ey.1 $176$ $2$ $2$ $2$
176.48.2.ez.1 $176$ $2$ $2$ $2$
184.48.1.iw.1 $184$ $2$ $2$ $1$
184.48.1.ja.1 $184$ $2$ $2$ $1$
184.48.1.jm.1 $184$ $2$ $2$ $1$
184.48.1.jq.1 $184$ $2$ $2$ $1$
208.48.1.fs.1 $208$ $2$ $2$ $1$
208.48.1.fu.1 $208$ $2$ $2$ $1$
208.48.1.ga.1 $208$ $2$ $2$ $1$
208.48.1.gb.1 $208$ $2$ $2$ $1$
208.48.1.gc.1 $208$ $2$ $2$ $1$
208.48.1.gd.1 $208$ $2$ $2$ $1$
208.48.1.gk.1 $208$ $2$ $2$ $1$
208.48.1.gm.1 $208$ $2$ $2$ $1$
208.48.2.fi.1 $208$ $2$ $2$ $2$
208.48.2.fj.1 $208$ $2$ $2$ $2$
232.48.1.iy.1 $232$ $2$ $2$ $1$
232.48.1.jc.1 $232$ $2$ $2$ $1$
232.48.1.jo.1 $232$ $2$ $2$ $1$
232.48.1.js.1 $232$ $2$ $2$ $1$
240.48.1.sq.1 $240$ $2$ $2$ $1$
240.48.1.ss.1 $240$ $2$ $2$ $1$
240.48.1.sy.1 $240$ $2$ $2$ $1$
240.48.1.sz.1 $240$ $2$ $2$ $1$
240.48.1.ta.1 $240$ $2$ $2$ $1$
240.48.1.tb.1 $240$ $2$ $2$ $1$
240.48.1.ti.1 $240$ $2$ $2$ $1$
240.48.1.tk.1 $240$ $2$ $2$ $1$
240.48.2.oo.1 $240$ $2$ $2$ $2$
240.48.2.op.1 $240$ $2$ $2$ $2$
248.48.1.iw.1 $248$ $2$ $2$ $1$
248.48.1.ja.1 $248$ $2$ $2$ $1$
248.48.1.jm.1 $248$ $2$ $2$ $1$
248.48.1.jq.1 $248$ $2$ $2$ $1$
264.48.1.cqo.1 $264$ $2$ $2$ $1$
264.48.1.cqs.1 $264$ $2$ $2$ $1$
264.48.1.cru.1 $264$ $2$ $2$ $1$
264.48.1.cry.1 $264$ $2$ $2$ $1$
272.48.1.fs.1 $272$ $2$ $2$ $1$
272.48.1.fu.1 $272$ $2$ $2$ $1$
272.48.1.ga.1 $272$ $2$ $2$ $1$
272.48.1.gb.1 $272$ $2$ $2$ $1$
272.48.1.gc.1 $272$ $2$ $2$ $1$
272.48.1.gd.1 $272$ $2$ $2$ $1$
272.48.1.gk.1 $272$ $2$ $2$ $1$
272.48.1.gm.1 $272$ $2$ $2$ $1$
272.48.2.fi.1 $272$ $2$ $2$ $2$
272.48.2.fj.1 $272$ $2$ $2$ $2$
280.48.1.cku.1 $280$ $2$ $2$ $1$
280.48.1.cky.1 $280$ $2$ $2$ $1$
280.48.1.cma.1 $280$ $2$ $2$ $1$
280.48.1.cme.1 $280$ $2$ $2$ $1$
296.48.1.iy.1 $296$ $2$ $2$ $1$
296.48.1.jc.1 $296$ $2$ $2$ $1$
296.48.1.jo.1 $296$ $2$ $2$ $1$
296.48.1.js.1 $296$ $2$ $2$ $1$
304.48.1.fq.1 $304$ $2$ $2$ $1$
304.48.1.fs.1 $304$ $2$ $2$ $1$
304.48.1.fy.1 $304$ $2$ $2$ $1$
304.48.1.fz.1 $304$ $2$ $2$ $1$
304.48.1.ga.1 $304$ $2$ $2$ $1$
304.48.1.gb.1 $304$ $2$ $2$ $1$
304.48.1.gi.1 $304$ $2$ $2$ $1$
304.48.1.gk.1 $304$ $2$ $2$ $1$
304.48.2.ey.1 $304$ $2$ $2$ $2$
304.48.2.ez.1 $304$ $2$ $2$ $2$
312.48.1.cqq.1 $312$ $2$ $2$ $1$
312.48.1.cqu.1 $312$ $2$ $2$ $1$
312.48.1.crw.1 $312$ $2$ $2$ $1$
312.48.1.csa.1 $312$ $2$ $2$ $1$
328.48.1.iy.1 $328$ $2$ $2$ $1$
328.48.1.jc.1 $328$ $2$ $2$ $1$
328.48.1.jo.1 $328$ $2$ $2$ $1$
328.48.1.js.1 $328$ $2$ $2$ $1$