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Properties

Label 8.24.1.bb.1
Level $8$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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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: $4$ (of which $2$ are rational) Cusp widths $4^{2}\cdot8^{2}$ Cusp orbits $1^{2}\cdot2$
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: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 8C1
Rouse and Zureick-Brown (RZB) label: X136
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.24.1.14

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&0\\6&7\end{bmatrix}$, $\begin{bmatrix}3&0\\2&5\end{bmatrix}$, $\begin{bmatrix}7&1\\2&5\end{bmatrix}$, $\begin{bmatrix}7&5\\6&5\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^3:D_4$
Contains $-I$: yes
Quadratic refinements: 8.48.1-8.bb.1.1, 8.48.1-8.bb.1.2, 16.48.1-8.bb.1.1, 16.48.1-8.bb.1.2, 16.48.1-8.bb.1.3, 16.48.1-8.bb.1.4, 24.48.1-8.bb.1.1, 24.48.1-8.bb.1.2, 40.48.1-8.bb.1.1, 40.48.1-8.bb.1.2, 48.48.1-8.bb.1.1, 48.48.1-8.bb.1.2, 48.48.1-8.bb.1.3, 48.48.1-8.bb.1.4, 56.48.1-8.bb.1.1, 56.48.1-8.bb.1.2, 80.48.1-8.bb.1.1, 80.48.1-8.bb.1.2, 80.48.1-8.bb.1.3, 80.48.1-8.bb.1.4, 88.48.1-8.bb.1.1, 88.48.1-8.bb.1.2, 104.48.1-8.bb.1.1, 104.48.1-8.bb.1.2, 112.48.1-8.bb.1.1, 112.48.1-8.bb.1.2, 112.48.1-8.bb.1.3, 112.48.1-8.bb.1.4, 120.48.1-8.bb.1.1, 120.48.1-8.bb.1.2, 136.48.1-8.bb.1.1, 136.48.1-8.bb.1.2, 152.48.1-8.bb.1.1, 152.48.1-8.bb.1.2, 168.48.1-8.bb.1.1, 168.48.1-8.bb.1.2, 176.48.1-8.bb.1.1, 176.48.1-8.bb.1.2, 176.48.1-8.bb.1.3, 176.48.1-8.bb.1.4, 184.48.1-8.bb.1.1, 184.48.1-8.bb.1.2, 208.48.1-8.bb.1.1, 208.48.1-8.bb.1.2, 208.48.1-8.bb.1.3, 208.48.1-8.bb.1.4, 232.48.1-8.bb.1.1, 232.48.1-8.bb.1.2, 240.48.1-8.bb.1.1, 240.48.1-8.bb.1.2, 240.48.1-8.bb.1.3, 240.48.1-8.bb.1.4, 248.48.1-8.bb.1.1, 248.48.1-8.bb.1.2, 264.48.1-8.bb.1.1, 264.48.1-8.bb.1.2, 272.48.1-8.bb.1.1, 272.48.1-8.bb.1.2, 272.48.1-8.bb.1.3, 272.48.1-8.bb.1.4, 280.48.1-8.bb.1.1, 280.48.1-8.bb.1.2, 296.48.1-8.bb.1.1, 296.48.1-8.bb.1.2, 304.48.1-8.bb.1.1, 304.48.1-8.bb.1.2, 304.48.1-8.bb.1.3, 304.48.1-8.bb.1.4, 312.48.1-8.bb.1.1, 312.48.1-8.bb.1.2, 328.48.1-8.bb.1.1, 328.48.1-8.bb.1.2
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 $
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Rational points

This modular curve has 2 rational cusps and 1 rational CM point, but no 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 $j$-heightWeierstrass model
no$\infty$ $0.000$$(4:0:1)$, $(0:1:0)$
32.a3 $-4$$1728$ $= 2^{6} \cdot 3^{3}$$7.455$$(6:-8:1)$, $(6:8:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{24x^{2}y^{6}-44592x^{2}y^{4}z^{2}+18137088x^{2}y^{2}z^{4}-2103903232x^{2}z^{6}-392xy^{6}z+457728xy^{4}z^{3}-154829056xy^{2}z^{5}+16109268992xz^{7}-y^{8}+4224y^{6}z^{2}-2556160y^{4}z^{4}+509605888y^{2}z^{6}-30774628352z^{8}}{z^{2}(x^{2}y^{4}-3776x^{2}y^{2}z^{2}+1217536x^{2}z^{4}-24xy^{4}z+38144xy^{2}z^{3}-9322496xz^{5}+336y^{4}z^{2}-196608y^{2}z^{4}+17809408z^{6})}$

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 Kernel decomposition
$X_{\mathrm{sp}}^+(4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
8.12.0.v.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.g.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.u.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.bm.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.bp.1 $8$ $2$ $2$ $1$ $0$ dimension zero
16.48.2.r.1 $16$ $2$ $2$ $2$ $0$ $1$
16.48.2.t.1 $16$ $2$ $2$ $2$ $0$ $1$
16.48.2.bp.1 $16$ $2$ $2$ $2$ $1$ $1$
16.48.2.br.1 $16$ $2$ $2$ $2$ $1$ $1$
24.48.1.hb.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.hf.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.hr.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.hv.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.5.gp.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
24.96.5.df.1 $24$ $4$ $4$ $5$ $0$ $1^{4}$
40.48.1.gf.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.gj.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.gv.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.gz.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.120.9.cx.1 $40$ $5$ $5$ $9$ $3$ $1^{6}\cdot2$
40.144.9.fb.1 $40$ $6$ $6$ $9$ $1$ $1^{6}\cdot2$
40.240.17.pp.1 $40$ $10$ $10$ $17$ $6$ $1^{12}\cdot2^{2}$
48.48.2.z.1 $48$ $2$ $2$ $2$ $1$ $1$
48.48.2.bb.1 $48$ $2$ $2$ $2$ $1$ $1$
48.48.2.bp.1 $48$ $2$ $2$ $2$ $0$ $1$
48.48.2.br.1 $48$ $2$ $2$ $2$ $0$ $1$
56.48.1.gd.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.gh.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.gt.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.gx.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.192.13.df.1 $56$ $8$ $8$ $13$ $3$ $1^{12}$
56.504.37.gp.1 $56$ $21$ $21$ $37$ $14$ $1^{8}\cdot2^{12}\cdot4$
56.672.49.gp.1 $56$ $28$ $28$ $49$ $17$ $1^{20}\cdot2^{12}\cdot4$
80.48.2.bf.1 $80$ $2$ $2$ $2$ $?$ not computed
80.48.2.bh.1 $80$ $2$ $2$ $2$ $?$ not computed
80.48.2.bv.1 $80$ $2$ $2$ $2$ $?$ not computed
80.48.2.bx.1 $80$ $2$ $2$ $2$ $?$ not computed
88.48.1.gd.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.gh.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.gt.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.gx.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.288.21.df.1 $88$ $12$ $12$ $21$ $?$ not computed
104.48.1.gf.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.gj.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.gv.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.gz.1 $104$ $2$ $2$ $1$ $?$ dimension zero
112.48.2.z.1 $112$ $2$ $2$ $2$ $?$ not computed
112.48.2.bb.1 $112$ $2$ $2$ $2$ $?$ not computed
112.48.2.bp.1 $112$ $2$ $2$ $2$ $?$ not computed
112.48.2.br.1 $112$ $2$ $2$ $2$ $?$ not computed
120.48.1.xd.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.xh.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.yj.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.yn.1 $120$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.gf.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.gj.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.gv.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.gz.1 $136$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.gd.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.gh.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.gt.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.gx.1 $152$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.xb.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.xf.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.yh.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.yl.1 $168$ $2$ $2$ $1$ $?$ dimension zero
176.48.2.z.1 $176$ $2$ $2$ $2$ $?$ not computed
176.48.2.bb.1 $176$ $2$ $2$ $2$ $?$ not computed
176.48.2.bp.1 $176$ $2$ $2$ $2$ $?$ not computed
176.48.2.br.1 $176$ $2$ $2$ $2$ $?$ not computed
184.48.1.gd.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.gh.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.gt.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.gx.1 $184$ $2$ $2$ $1$ $?$ dimension zero
208.48.2.bf.1 $208$ $2$ $2$ $2$ $?$ not computed
208.48.2.bh.1 $208$ $2$ $2$ $2$ $?$ not computed
208.48.2.bv.1 $208$ $2$ $2$ $2$ $?$ not computed
208.48.2.bx.1 $208$ $2$ $2$ $2$ $?$ not computed
232.48.1.gf.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.gj.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.gv.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.gz.1 $232$ $2$ $2$ $1$ $?$ dimension zero
240.48.2.bf.1 $240$ $2$ $2$ $2$ $?$ not computed
240.48.2.bh.1 $240$ $2$ $2$ $2$ $?$ not computed
240.48.2.bv.1 $240$ $2$ $2$ $2$ $?$ not computed
240.48.2.bx.1 $240$ $2$ $2$ $2$ $?$ not computed
248.48.1.gd.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.gh.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.gt.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.gx.1 $248$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.xb.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.xf.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.yh.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.yl.1 $264$ $2$ $2$ $1$ $?$ dimension zero
272.48.2.bf.1 $272$ $2$ $2$ $2$ $?$ not computed
272.48.2.bh.1 $272$ $2$ $2$ $2$ $?$ not computed
272.48.2.bv.1 $272$ $2$ $2$ $2$ $?$ not computed
272.48.2.bx.1 $272$ $2$ $2$ $2$ $?$ not computed
280.48.1.wf.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.wj.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.xl.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.xp.1 $280$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.gf.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.gj.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.gv.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.gz.1 $296$ $2$ $2$ $1$ $?$ dimension zero
304.48.2.z.1 $304$ $2$ $2$ $2$ $?$ not computed
304.48.2.bb.1 $304$ $2$ $2$ $2$ $?$ not computed
304.48.2.bp.1 $304$ $2$ $2$ $2$ $?$ not computed
304.48.2.br.1 $304$ $2$ $2$ $2$ $?$ not computed
312.48.1.xd.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.xh.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.yj.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.yn.1 $312$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.gf.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.gj.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.gv.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.gz.1 $328$ $2$ $2$ $1$ $?$ dimension zero