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Properties

Label 8.12.1.b.1
Level $8$
Index $12$
Genus $1$
Analytic rank $0$
Cusps $2$
$\Q$-cusps $2$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $32$
Index: $12$ $\PSL_2$-index:$12$
Genus: $1 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$
Cusps: $2$ (all of which are rational) Cusp widths $4\cdot8$ Cusp orbits $1^{2}$
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 $\quad(D =$ $-8$)

Other labels

Cummins and Pauli (CP) label: 8A1
Rouse and Zureick-Brown (RZB) label: X51
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.12.1.1

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&3\\6&7\end{bmatrix}$, $\begin{bmatrix}1&7\\2&1\end{bmatrix}$, $\begin{bmatrix}5&4\\0&5\end{bmatrix}$, $\begin{bmatrix}7&1\\6&3\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^4.D_4$
Contains $-I$: yes
Quadratic refinements: 8.24.1-8.b.1.1, 8.24.1-8.b.1.2, 8.24.1-8.b.1.3, 8.24.1-8.b.1.4, 16.24.1-8.b.1.1, 16.24.1-8.b.1.2, 16.24.1-8.b.1.3, 16.24.1-8.b.1.4, 24.24.1-8.b.1.1, 24.24.1-8.b.1.2, 24.24.1-8.b.1.3, 24.24.1-8.b.1.4, 40.24.1-8.b.1.1, 40.24.1-8.b.1.2, 40.24.1-8.b.1.3, 40.24.1-8.b.1.4, 48.24.1-8.b.1.1, 48.24.1-8.b.1.2, 48.24.1-8.b.1.3, 48.24.1-8.b.1.4, 56.24.1-8.b.1.1, 56.24.1-8.b.1.2, 56.24.1-8.b.1.3, 56.24.1-8.b.1.4, 80.24.1-8.b.1.1, 80.24.1-8.b.1.2, 80.24.1-8.b.1.3, 80.24.1-8.b.1.4, 88.24.1-8.b.1.1, 88.24.1-8.b.1.2, 88.24.1-8.b.1.3, 88.24.1-8.b.1.4, 104.24.1-8.b.1.1, 104.24.1-8.b.1.2, 104.24.1-8.b.1.3, 104.24.1-8.b.1.4, 112.24.1-8.b.1.1, 112.24.1-8.b.1.2, 112.24.1-8.b.1.3, 112.24.1-8.b.1.4, 120.24.1-8.b.1.1, 120.24.1-8.b.1.2, 120.24.1-8.b.1.3, 120.24.1-8.b.1.4, 136.24.1-8.b.1.1, 136.24.1-8.b.1.2, 136.24.1-8.b.1.3, 136.24.1-8.b.1.4, 152.24.1-8.b.1.1, 152.24.1-8.b.1.2, 152.24.1-8.b.1.3, 152.24.1-8.b.1.4, 168.24.1-8.b.1.1, 168.24.1-8.b.1.2, 168.24.1-8.b.1.3, 168.24.1-8.b.1.4, 176.24.1-8.b.1.1, 176.24.1-8.b.1.2, 176.24.1-8.b.1.3, 176.24.1-8.b.1.4, 184.24.1-8.b.1.1, 184.24.1-8.b.1.2, 184.24.1-8.b.1.3, 184.24.1-8.b.1.4, 208.24.1-8.b.1.1, 208.24.1-8.b.1.2, 208.24.1-8.b.1.3, 208.24.1-8.b.1.4, 232.24.1-8.b.1.1, 232.24.1-8.b.1.2, 232.24.1-8.b.1.3, 232.24.1-8.b.1.4, 240.24.1-8.b.1.1, 240.24.1-8.b.1.2, 240.24.1-8.b.1.3, 240.24.1-8.b.1.4, 248.24.1-8.b.1.1, 248.24.1-8.b.1.2, 248.24.1-8.b.1.3, 248.24.1-8.b.1.4, 264.24.1-8.b.1.1, 264.24.1-8.b.1.2, 264.24.1-8.b.1.3, 264.24.1-8.b.1.4, 272.24.1-8.b.1.1, 272.24.1-8.b.1.2, 272.24.1-8.b.1.3, 272.24.1-8.b.1.4, 280.24.1-8.b.1.1, 280.24.1-8.b.1.2, 280.24.1-8.b.1.3, 280.24.1-8.b.1.4, 296.24.1-8.b.1.1, 296.24.1-8.b.1.2, 296.24.1-8.b.1.3, 296.24.1-8.b.1.4, 304.24.1-8.b.1.1, 304.24.1-8.b.1.2, 304.24.1-8.b.1.3, 304.24.1-8.b.1.4, 312.24.1-8.b.1.1, 312.24.1-8.b.1.2, 312.24.1-8.b.1.3, 312.24.1-8.b.1.4, 328.24.1-8.b.1.1, 328.24.1-8.b.1.2, 328.24.1-8.b.1.3, 328.24.1-8.b.1.4
Cyclic 8-isogeny field degree: $4$
Cyclic 8-torsion field degree: $16$
Full 8-torsion field degree: $128$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 4x $
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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$$(0:1:0)$, $(0:0:1)$
256.a1 $-8$$8000$ $= 2^{6} \cdot 5^{3}$$8.987$$(2:-4:1)$, $(2:4:1)$

Maps to other modular curves

$j$-invariant map of degree 12 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{7x^{2}z^{2}-5xy^{2}z+y^{4}+z^{4}}{z^{2}x^{2}}$

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
4.6.0.d.1 $4$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
8.24.1.c.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.g.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.i.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.l.1 $8$ $2$ $2$ $1$ $0$ dimension zero
16.24.2.c.1 $16$ $2$ $2$ $2$ $0$ $1$
16.24.2.d.1 $16$ $2$ $2$ $2$ $0$ $1$
16.24.2.e.1 $16$ $2$ $2$ $2$ $0$ $1$
16.24.2.f.1 $16$ $2$ $2$ $2$ $1$ $1$
24.24.1.s.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.t.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.w.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.x.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.36.3.b.1 $24$ $3$ $3$ $3$ $1$ $1^{2}$
24.48.3.b.1 $24$ $4$ $4$ $3$ $0$ $1^{2}$
40.24.1.s.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.t.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.w.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.x.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.60.5.b.1 $40$ $5$ $5$ $5$ $0$ $1^{2}\cdot2$
40.72.5.b.1 $40$ $6$ $6$ $5$ $0$ $1^{2}\cdot2$
40.120.9.dd.1 $40$ $10$ $10$ $9$ $1$ $1^{4}\cdot2^{2}$
48.24.2.a.1 $48$ $2$ $2$ $2$ $0$ $1$
48.24.2.b.1 $48$ $2$ $2$ $2$ $0$ $1$
48.24.2.c.1 $48$ $2$ $2$ $2$ $0$ $1$
48.24.2.d.1 $48$ $2$ $2$ $2$ $1$ $1$
56.24.1.s.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.24.1.t.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.24.1.w.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.24.1.x.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.7.b.1 $56$ $8$ $8$ $7$ $1$ $1^{6}$
56.252.19.b.1 $56$ $21$ $21$ $19$ $5$ $1^{2}\cdot2^{6}\cdot4$
56.336.25.b.1 $56$ $28$ $28$ $25$ $6$ $1^{8}\cdot2^{6}\cdot4$
80.24.2.a.1 $80$ $2$ $2$ $2$ $?$ not computed
80.24.2.b.1 $80$ $2$ $2$ $2$ $?$ not computed
80.24.2.c.1 $80$ $2$ $2$ $2$ $?$ not computed
80.24.2.d.1 $80$ $2$ $2$ $2$ $?$ not computed
88.24.1.s.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.24.1.t.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.24.1.w.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.24.1.x.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.144.11.b.1 $88$ $12$ $12$ $11$ $?$ not computed
104.24.1.s.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.24.1.t.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.24.1.w.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.24.1.x.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.168.13.b.1 $104$ $14$ $14$ $13$ $?$ not computed
112.24.2.a.1 $112$ $2$ $2$ $2$ $?$ not computed
112.24.2.b.1 $112$ $2$ $2$ $2$ $?$ not computed
112.24.2.c.1 $112$ $2$ $2$ $2$ $?$ not computed
112.24.2.d.1 $112$ $2$ $2$ $2$ $?$ not computed
120.24.1.s.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.24.1.t.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.24.1.w.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.24.1.x.1 $120$ $2$ $2$ $1$ $?$ dimension zero
136.24.1.s.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.24.1.t.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.24.1.w.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.24.1.x.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.216.17.b.1 $136$ $18$ $18$ $17$ $?$ not computed
152.24.1.s.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.24.1.t.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.24.1.w.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.24.1.x.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.240.19.b.1 $152$ $20$ $20$ $19$ $?$ not computed
168.24.1.s.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.24.1.t.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.24.1.w.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.24.1.x.1 $168$ $2$ $2$ $1$ $?$ dimension zero
176.24.2.a.1 $176$ $2$ $2$ $2$ $?$ not computed
176.24.2.b.1 $176$ $2$ $2$ $2$ $?$ not computed
176.24.2.c.1 $176$ $2$ $2$ $2$ $?$ not computed
176.24.2.d.1 $176$ $2$ $2$ $2$ $?$ not computed
184.24.1.s.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.24.1.t.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.24.1.w.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.24.1.x.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.288.23.b.1 $184$ $24$ $24$ $23$ $?$ not computed
208.24.2.a.1 $208$ $2$ $2$ $2$ $?$ not computed
208.24.2.b.1 $208$ $2$ $2$ $2$ $?$ not computed
208.24.2.c.1 $208$ $2$ $2$ $2$ $?$ not computed
208.24.2.d.1 $208$ $2$ $2$ $2$ $?$ not computed
232.24.1.s.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.24.1.t.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.24.1.w.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.24.1.x.1 $232$ $2$ $2$ $1$ $?$ dimension zero
240.24.2.a.1 $240$ $2$ $2$ $2$ $?$ not computed
240.24.2.b.1 $240$ $2$ $2$ $2$ $?$ not computed
240.24.2.c.1 $240$ $2$ $2$ $2$ $?$ not computed
240.24.2.d.1 $240$ $2$ $2$ $2$ $?$ not computed
248.24.1.s.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.24.1.t.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.24.1.w.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.24.1.x.1 $248$ $2$ $2$ $1$ $?$ dimension zero
264.24.1.s.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.24.1.t.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.24.1.w.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.24.1.x.1 $264$ $2$ $2$ $1$ $?$ dimension zero
272.24.2.a.1 $272$ $2$ $2$ $2$ $?$ not computed
272.24.2.b.1 $272$ $2$ $2$ $2$ $?$ not computed
272.24.2.c.1 $272$ $2$ $2$ $2$ $?$ not computed
272.24.2.d.1 $272$ $2$ $2$ $2$ $?$ not computed
280.24.1.s.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.24.1.t.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.24.1.w.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.24.1.x.1 $280$ $2$ $2$ $1$ $?$ dimension zero
296.24.1.s.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.24.1.t.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.24.1.w.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.24.1.x.1 $296$ $2$ $2$ $1$ $?$ dimension zero
304.24.2.a.1 $304$ $2$ $2$ $2$ $?$ not computed
304.24.2.b.1 $304$ $2$ $2$ $2$ $?$ not computed
304.24.2.c.1 $304$ $2$ $2$ $2$ $?$ not computed
304.24.2.d.1 $304$ $2$ $2$ $2$ $?$ not computed
312.24.1.s.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.24.1.t.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.24.1.w.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.24.1.x.1 $312$ $2$ $2$ $1$ $?$ dimension zero
328.24.1.s.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.24.1.t.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.24.1.w.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.24.1.x.1 $328$ $2$ $2$ $1$ $?$ dimension zero