Modular curve 8.24.1.y.1

• Label: 8.24.1.y.1
• Level: $8$
• Index: $24$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $4$
• $\Q$-cusps: $0$

Invariants

Level:	$8$		$\SL_2$-level:	$8$		Newform level:	$64$
Index:	$24$		$\PSL_2$-index:	$24$
Genus:	$1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (none of which are rational)		Cusp widths	$4^{2}\cdot8^{2}$		Cusp orbits	$2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label:	8C1
Rouse and Zureick-Brown (RZB) label:	X128
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	8.24.1.28

Level structure

$\GL_2(\Z/8\Z)$-generators:	$\begin{bmatrix}1&5\\2&3\end{bmatrix}$, $\begin{bmatrix}5&1\\4&7\end{bmatrix}$, $\begin{bmatrix}7&6\\0&3\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup:	$C_2^3.C_2^3$
Contains $-I$:	yes
Quadratic refinements:	16.48.1-8.y.1.1, 16.48.1-8.y.1.2, 16.48.1-8.y.1.3, 16.48.1-8.y.1.4, 16.48.1-8.y.1.5, 16.48.1-8.y.1.6, 48.48.1-8.y.1.1, 48.48.1-8.y.1.2, 48.48.1-8.y.1.3, 48.48.1-8.y.1.4, 48.48.1-8.y.1.5, 48.48.1-8.y.1.6, 80.48.1-8.y.1.1, 80.48.1-8.y.1.2, 80.48.1-8.y.1.3, 80.48.1-8.y.1.4, 80.48.1-8.y.1.5, 80.48.1-8.y.1.6, 112.48.1-8.y.1.1, 112.48.1-8.y.1.2, 112.48.1-8.y.1.3, 112.48.1-8.y.1.4, 112.48.1-8.y.1.5, 112.48.1-8.y.1.6, 176.48.1-8.y.1.1, 176.48.1-8.y.1.2, 176.48.1-8.y.1.3, 176.48.1-8.y.1.4, 176.48.1-8.y.1.5, 176.48.1-8.y.1.6, 208.48.1-8.y.1.1, 208.48.1-8.y.1.2, 208.48.1-8.y.1.3, 208.48.1-8.y.1.4, 208.48.1-8.y.1.5, 208.48.1-8.y.1.6, 240.48.1-8.y.1.1, 240.48.1-8.y.1.2, 240.48.1-8.y.1.3, 240.48.1-8.y.1.4, 240.48.1-8.y.1.5, 240.48.1-8.y.1.6, 272.48.1-8.y.1.1, 272.48.1-8.y.1.2, 272.48.1-8.y.1.3, 272.48.1-8.y.1.4, 272.48.1-8.y.1.5, 272.48.1-8.y.1.6, 304.48.1-8.y.1.1, 304.48.1-8.y.1.2, 304.48.1-8.y.1.3, 304.48.1-8.y.1.4, 304.48.1-8.y.1.5, 304.48.1-8.y.1.6
Cyclic 8-isogeny field degree:	$4$
Cyclic 8-torsion field degree:	$16$
Full 8-torsion field degree:	$64$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $

$=$

$ x^{3} - 44x - 112 $

Rational points

This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve	CM	$j$-invariant		$j$-height	Weierstrass model
32.a3	$-4$	$1728$	$= 2^{6} \cdot 3^{3}$	$7.455$	$(-4:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle 2^6\,\frac{2808x^{2}y^{6}+60185424x^{2}y^{4}z^{2}+82141387776x^{2}y^{2}z^{4}+20426864487424x^{2}z^{6}+126504xy^{6}z+879177216xy^{4}z^{3}+783830985472xy^{2}z^{5}+156405528940544xz^{7}+27y^{8}+3287168y^{6}z^{2}+8460513280y^{4}z^{4}+3573408864256y^{2}z^{6}+298792284073984z^{8}}{8x^{2}y^{6}-272x^{2}y^{4}z^{2}-1024x^{2}y^{2}z^{4}+1024x^{2}z^{6}-8xy^{6}z+1024xy^{4}z^{3}+4352xy^{2}z^{5}-4096xz^{7}+y^{8}-384y^{6}z^{2}+8192y^{4}z^{4}+26624y^{2}z^{6}-28672z^{8}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
8.12.0.s.1	$8$	$2$	$2$	$0$	$0$	full Jacobian
8.12.0.u.1	$8$	$2$	$2$	$0$	$0$	full Jacobian
8.12.1.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
8.48.1.d.1	$8$	$2$	$2$	$1$	$0$	dimension zero
8.48.1.q.1	$8$	$2$	$2$	$1$	$0$	dimension zero
8.48.1.bl.1	$8$	$2$	$2$	$1$	$0$	dimension zero
8.48.1.bp.1	$8$	$2$	$2$	$1$	$0$	dimension zero
16.48.2.bg.1	$16$	$2$	$2$	$2$	$1$	$1$
16.48.2.bh.1	$16$	$2$	$2$	$2$	$0$	$1$
16.48.2.bi.1	$16$	$2$	$2$	$2$	$1$	$1$
16.48.2.bj.1	$16$	$2$	$2$	$2$	$0$	$1$
24.48.1.gy.1	$24$	$2$	$2$	$1$	$0$	dimension zero
24.48.1.hc.1	$24$	$2$	$2$	$1$	$0$	dimension zero
24.48.1.ho.1	$24$	$2$	$2$	$1$	$0$	dimension zero
24.48.1.hs.1	$24$	$2$	$2$	$1$	$0$	dimension zero
24.72.5.gm.1	$24$	$3$	$3$	$5$	$1$	$1^{4}$
24.96.5.dc.1	$24$	$4$	$4$	$5$	$0$	$1^{4}$
40.48.1.gc.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.48.1.gg.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.48.1.gs.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.48.1.gw.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.120.9.cu.1	$40$	$5$	$5$	$9$	$4$	$1^{6}\cdot2$
40.144.9.ey.1	$40$	$6$	$6$	$9$	$1$	$1^{6}\cdot2$
40.240.17.pm.1	$40$	$10$	$10$	$17$	$8$	$1^{12}\cdot2^{2}$
48.48.2.bg.1	$48$	$2$	$2$	$2$	$0$	$1$
48.48.2.bh.1	$48$	$2$	$2$	$2$	$1$	$1$
48.48.2.bi.1	$48$	$2$	$2$	$2$	$0$	$1$
48.48.2.bj.1	$48$	$2$	$2$	$2$	$1$	$1$
56.48.1.ga.1	$56$	$2$	$2$	$1$	$0$	dimension zero
56.48.1.ge.1	$56$	$2$	$2$	$1$	$0$	dimension zero
56.48.1.gq.1	$56$	$2$	$2$	$1$	$0$	dimension zero
56.48.1.gu.1	$56$	$2$	$2$	$1$	$0$	dimension zero
56.192.13.dc.1	$56$	$8$	$8$	$13$	$4$	$1^{12}$
56.504.37.gm.1	$56$	$21$	$21$	$37$	$17$	$1^{8}\cdot2^{12}\cdot4$
56.672.49.gm.1	$56$	$28$	$28$	$49$	$21$	$1^{20}\cdot2^{12}\cdot4$
80.48.2.bm.1	$80$	$2$	$2$	$2$	$?$	not computed
80.48.2.bn.1	$80$	$2$	$2$	$2$	$?$	not computed
80.48.2.bo.1	$80$	$2$	$2$	$2$	$?$	not computed
80.48.2.bp.1	$80$	$2$	$2$	$2$	$?$	not computed
88.48.1.ga.1	$88$	$2$	$2$	$1$	$?$	dimension zero
88.48.1.ge.1	$88$	$2$	$2$	$1$	$?$	dimension zero
88.48.1.gq.1	$88$	$2$	$2$	$1$	$?$	dimension zero
88.48.1.gu.1	$88$	$2$	$2$	$1$	$?$	dimension zero
88.288.21.dc.1	$88$	$12$	$12$	$21$	$?$	not computed
104.48.1.gc.1	$104$	$2$	$2$	$1$	$?$	dimension zero
104.48.1.gg.1	$104$	$2$	$2$	$1$	$?$	dimension zero
104.48.1.gs.1	$104$	$2$	$2$	$1$	$?$	dimension zero
104.48.1.gw.1	$104$	$2$	$2$	$1$	$?$	dimension zero
112.48.2.bg.1	$112$	$2$	$2$	$2$	$?$	not computed
112.48.2.bh.1	$112$	$2$	$2$	$2$	$?$	not computed
112.48.2.bi.1	$112$	$2$	$2$	$2$	$?$	not computed
112.48.2.bj.1	$112$	$2$	$2$	$2$	$?$	not computed
120.48.1.xa.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.48.1.xe.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.48.1.yg.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.48.1.yk.1	$120$	$2$	$2$	$1$	$?$	dimension zero
136.48.1.gc.1	$136$	$2$	$2$	$1$	$?$	dimension zero
136.48.1.gg.1	$136$	$2$	$2$	$1$	$?$	dimension zero
136.48.1.gs.1	$136$	$2$	$2$	$1$	$?$	dimension zero
136.48.1.gw.1	$136$	$2$	$2$	$1$	$?$	dimension zero
152.48.1.ga.1	$152$	$2$	$2$	$1$	$?$	dimension zero
152.48.1.ge.1	$152$	$2$	$2$	$1$	$?$	dimension zero
152.48.1.gq.1	$152$	$2$	$2$	$1$	$?$	dimension zero
152.48.1.gu.1	$152$	$2$	$2$	$1$	$?$	dimension zero
168.48.1.wy.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.48.1.xc.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.48.1.ye.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.48.1.yi.1	$168$	$2$	$2$	$1$	$?$	dimension zero
176.48.2.bg.1	$176$	$2$	$2$	$2$	$?$	not computed
176.48.2.bh.1	$176$	$2$	$2$	$2$	$?$	not computed
176.48.2.bi.1	$176$	$2$	$2$	$2$	$?$	not computed
176.48.2.bj.1	$176$	$2$	$2$	$2$	$?$	not computed
184.48.1.ga.1	$184$	$2$	$2$	$1$	$?$	dimension zero
184.48.1.ge.1	$184$	$2$	$2$	$1$	$?$	dimension zero
184.48.1.gq.1	$184$	$2$	$2$	$1$	$?$	dimension zero
184.48.1.gu.1	$184$	$2$	$2$	$1$	$?$	dimension zero
208.48.2.bm.1	$208$	$2$	$2$	$2$	$?$	not computed
208.48.2.bn.1	$208$	$2$	$2$	$2$	$?$	not computed
208.48.2.bo.1	$208$	$2$	$2$	$2$	$?$	not computed
208.48.2.bp.1	$208$	$2$	$2$	$2$	$?$	not computed
232.48.1.gc.1	$232$	$2$	$2$	$1$	$?$	dimension zero
232.48.1.gg.1	$232$	$2$	$2$	$1$	$?$	dimension zero
232.48.1.gs.1	$232$	$2$	$2$	$1$	$?$	dimension zero
232.48.1.gw.1	$232$	$2$	$2$	$1$	$?$	dimension zero
240.48.2.bm.1	$240$	$2$	$2$	$2$	$?$	not computed
240.48.2.bn.1	$240$	$2$	$2$	$2$	$?$	not computed
240.48.2.bo.1	$240$	$2$	$2$	$2$	$?$	not computed
240.48.2.bp.1	$240$	$2$	$2$	$2$	$?$	not computed
248.48.1.ga.1	$248$	$2$	$2$	$1$	$?$	dimension zero
248.48.1.ge.1	$248$	$2$	$2$	$1$	$?$	dimension zero
248.48.1.gq.1	$248$	$2$	$2$	$1$	$?$	dimension zero
248.48.1.gu.1	$248$	$2$	$2$	$1$	$?$	dimension zero
264.48.1.wy.1	$264$	$2$	$2$	$1$	$?$	dimension zero
264.48.1.xc.1	$264$	$2$	$2$	$1$	$?$	dimension zero
264.48.1.ye.1	$264$	$2$	$2$	$1$	$?$	dimension zero
264.48.1.yi.1	$264$	$2$	$2$	$1$	$?$	dimension zero
272.48.2.bm.1	$272$	$2$	$2$	$2$	$?$	not computed
272.48.2.bn.1	$272$	$2$	$2$	$2$	$?$	not computed
272.48.2.bo.1	$272$	$2$	$2$	$2$	$?$	not computed
272.48.2.bp.1	$272$	$2$	$2$	$2$	$?$	not computed
280.48.1.wc.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.48.1.wg.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.48.1.xi.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.48.1.xm.1	$280$	$2$	$2$	$1$	$?$	dimension zero
296.48.1.gc.1	$296$	$2$	$2$	$1$	$?$	dimension zero
296.48.1.gg.1	$296$	$2$	$2$	$1$	$?$	dimension zero
296.48.1.gs.1	$296$	$2$	$2$	$1$	$?$	dimension zero
296.48.1.gw.1	$296$	$2$	$2$	$1$	$?$	dimension zero
304.48.2.bg.1	$304$	$2$	$2$	$2$	$?$	not computed
304.48.2.bh.1	$304$	$2$	$2$	$2$	$?$	not computed
304.48.2.bi.1	$304$	$2$	$2$	$2$	$?$	not computed
304.48.2.bj.1	$304$	$2$	$2$	$2$	$?$	not computed
312.48.1.xa.1	$312$	$2$	$2$	$1$	$?$	dimension zero
312.48.1.xe.1	$312$	$2$	$2$	$1$	$?$	dimension zero
312.48.1.yg.1	$312$	$2$	$2$	$1$	$?$	dimension zero
312.48.1.yk.1	$312$	$2$	$2$	$1$	$?$	dimension zero
328.48.1.gc.1	$328$	$2$	$2$	$1$	$?$	dimension zero
328.48.1.gg.1	$328$	$2$	$2$	$1$	$?$	dimension zero
328.48.1.gs.1	$328$	$2$	$2$	$1$	$?$	dimension zero
328.48.1.gw.1	$328$	$2$	$2$	$1$	$?$	dimension zero