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Properties

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

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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: $4$ (all of which are rational) Cusp widths $4^{2}\cdot8^{2}$ Cusp orbits $1^{4}$
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: 8B1
Rouse and Zureick-Brown (RZB) label: X129
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.24.1.2

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}3&2\\0&7\end{bmatrix}$, $\begin{bmatrix}3&6\\0&1\end{bmatrix}$, $\begin{bmatrix}5&6\\4&5\end{bmatrix}$, $\begin{bmatrix}7&6\\0&5\end{bmatrix}$, $\begin{bmatrix}7&6\\4&1\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $D_4\times C_2^3$
Contains $-I$: yes
Quadratic refinements: 8.48.1-8.d.1.1, 8.48.1-8.d.1.2, 8.48.1-8.d.1.3, 8.48.1-8.d.1.4, 8.48.1-8.d.1.5, 8.48.1-8.d.1.6, 8.48.1-8.d.1.7, 8.48.1-8.d.1.8, 8.48.1-8.d.1.9, 8.48.1-8.d.1.10, 24.48.1-8.d.1.1, 24.48.1-8.d.1.2, 24.48.1-8.d.1.3, 24.48.1-8.d.1.4, 24.48.1-8.d.1.5, 24.48.1-8.d.1.6, 24.48.1-8.d.1.7, 24.48.1-8.d.1.8, 24.48.1-8.d.1.9, 24.48.1-8.d.1.10, 40.48.1-8.d.1.1, 40.48.1-8.d.1.2, 40.48.1-8.d.1.3, 40.48.1-8.d.1.4, 40.48.1-8.d.1.5, 40.48.1-8.d.1.6, 40.48.1-8.d.1.7, 40.48.1-8.d.1.8, 40.48.1-8.d.1.9, 40.48.1-8.d.1.10, 56.48.1-8.d.1.1, 56.48.1-8.d.1.2, 56.48.1-8.d.1.3, 56.48.1-8.d.1.4, 56.48.1-8.d.1.5, 56.48.1-8.d.1.6, 56.48.1-8.d.1.7, 56.48.1-8.d.1.8, 56.48.1-8.d.1.9, 56.48.1-8.d.1.10, 88.48.1-8.d.1.1, 88.48.1-8.d.1.2, 88.48.1-8.d.1.3, 88.48.1-8.d.1.4, 88.48.1-8.d.1.5, 88.48.1-8.d.1.6, 88.48.1-8.d.1.7, 88.48.1-8.d.1.8, 88.48.1-8.d.1.9, 88.48.1-8.d.1.10, 104.48.1-8.d.1.1, 104.48.1-8.d.1.2, 104.48.1-8.d.1.3, 104.48.1-8.d.1.4, 104.48.1-8.d.1.5, 104.48.1-8.d.1.6, 104.48.1-8.d.1.7, 104.48.1-8.d.1.8, 104.48.1-8.d.1.9, 104.48.1-8.d.1.10, 120.48.1-8.d.1.1, 120.48.1-8.d.1.2, 120.48.1-8.d.1.3, 120.48.1-8.d.1.4, 120.48.1-8.d.1.5, 120.48.1-8.d.1.6, 120.48.1-8.d.1.7, 120.48.1-8.d.1.8, 120.48.1-8.d.1.9, 120.48.1-8.d.1.10, 136.48.1-8.d.1.1, 136.48.1-8.d.1.2, 136.48.1-8.d.1.3, 136.48.1-8.d.1.4, 136.48.1-8.d.1.5, 136.48.1-8.d.1.6, 136.48.1-8.d.1.7, 136.48.1-8.d.1.8, 136.48.1-8.d.1.9, 136.48.1-8.d.1.10, 152.48.1-8.d.1.1, 152.48.1-8.d.1.2, 152.48.1-8.d.1.3, 152.48.1-8.d.1.4, 152.48.1-8.d.1.5, 152.48.1-8.d.1.6, 152.48.1-8.d.1.7, 152.48.1-8.d.1.8, 152.48.1-8.d.1.9, 152.48.1-8.d.1.10, 168.48.1-8.d.1.1, 168.48.1-8.d.1.2, 168.48.1-8.d.1.3, 168.48.1-8.d.1.4, 168.48.1-8.d.1.5, 168.48.1-8.d.1.6, 168.48.1-8.d.1.7, 168.48.1-8.d.1.8, 168.48.1-8.d.1.9, 168.48.1-8.d.1.10, 184.48.1-8.d.1.1, 184.48.1-8.d.1.2, 184.48.1-8.d.1.3, 184.48.1-8.d.1.4, 184.48.1-8.d.1.5, 184.48.1-8.d.1.6, 184.48.1-8.d.1.7, 184.48.1-8.d.1.8, 184.48.1-8.d.1.9, 184.48.1-8.d.1.10, 232.48.1-8.d.1.1, 232.48.1-8.d.1.2, 232.48.1-8.d.1.3, 232.48.1-8.d.1.4, 232.48.1-8.d.1.5, 232.48.1-8.d.1.6, 232.48.1-8.d.1.7, 232.48.1-8.d.1.8, 232.48.1-8.d.1.9, 232.48.1-8.d.1.10, 248.48.1-8.d.1.1, 248.48.1-8.d.1.2, 248.48.1-8.d.1.3, 248.48.1-8.d.1.4, 248.48.1-8.d.1.5, 248.48.1-8.d.1.6, 248.48.1-8.d.1.7, 248.48.1-8.d.1.8, 248.48.1-8.d.1.9, 248.48.1-8.d.1.10, 264.48.1-8.d.1.1, 264.48.1-8.d.1.2, 264.48.1-8.d.1.3, 264.48.1-8.d.1.4, 264.48.1-8.d.1.5, 264.48.1-8.d.1.6, 264.48.1-8.d.1.7, 264.48.1-8.d.1.8, 264.48.1-8.d.1.9, 264.48.1-8.d.1.10, 280.48.1-8.d.1.1, 280.48.1-8.d.1.2, 280.48.1-8.d.1.3, 280.48.1-8.d.1.4, 280.48.1-8.d.1.5, 280.48.1-8.d.1.6, 280.48.1-8.d.1.7, 280.48.1-8.d.1.8, 280.48.1-8.d.1.9, 280.48.1-8.d.1.10, 296.48.1-8.d.1.1, 296.48.1-8.d.1.2, 296.48.1-8.d.1.3, 296.48.1-8.d.1.4, 296.48.1-8.d.1.5, 296.48.1-8.d.1.6, 296.48.1-8.d.1.7, 296.48.1-8.d.1.8, 296.48.1-8.d.1.9, 296.48.1-8.d.1.10, 312.48.1-8.d.1.1, 312.48.1-8.d.1.2, 312.48.1-8.d.1.3, 312.48.1-8.d.1.4, 312.48.1-8.d.1.5, 312.48.1-8.d.1.6, 312.48.1-8.d.1.7, 312.48.1-8.d.1.8, 312.48.1-8.d.1.9, 312.48.1-8.d.1.10, 328.48.1-8.d.1.1, 328.48.1-8.d.1.2, 328.48.1-8.d.1.3, 328.48.1-8.d.1.4, 328.48.1-8.d.1.5, 328.48.1-8.d.1.6, 328.48.1-8.d.1.7, 328.48.1-8.d.1.8, 328.48.1-8.d.1.9, 328.48.1-8.d.1.10
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $8$
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} - 4x $
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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
$(0:1:0)$, $(2:0:1)$, $(0:0:1)$, $(-2:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{48x^{2}y^{4}z^{2}+4xy^{6}z+768xy^{2}z^{5}+y^{8}+4096z^{8}}{z^{2}y^{4}x^{2}}$

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
$X_{\pm1}(2,4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
8.12.0.x.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.12.1.a.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.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.l.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.m.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.m.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.n.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.p.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.u.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.w.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.x.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.x.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.y.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.ba.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.5.d.1 $24$ $3$ $3$ $5$ $0$ $1^{4}$
24.96.5.d.1 $24$ $4$ $4$ $5$ $1$ $1^{4}$
40.48.1.u.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.w.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.x.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.x.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.y.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.ba.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.120.9.d.1 $40$ $5$ $5$ $9$ $2$ $1^{6}\cdot2$
40.144.9.d.1 $40$ $6$ $6$ $9$ $0$ $1^{6}\cdot2$
40.240.17.bv.1 $40$ $10$ $10$ $17$ $4$ $1^{12}\cdot2^{2}$
56.48.1.u.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.w.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.x.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.x.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.y.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.ba.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.192.13.d.1 $56$ $8$ $8$ $13$ $3$ $1^{8}\cdot2^{2}$
56.504.37.d.1 $56$ $21$ $21$ $37$ $9$ $1^{4}\cdot2^{14}\cdot4$
56.672.49.d.1 $56$ $28$ $28$ $49$ $12$ $1^{12}\cdot2^{16}\cdot4$
88.48.1.u.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.w.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.x.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.x.2 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.y.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.ba.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.288.21.d.1 $88$ $12$ $12$ $21$ $?$ not computed
104.48.1.u.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.w.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.x.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.x.2 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.y.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.ba.1 $104$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.u.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.w.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.x.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.x.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.y.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.ba.1 $120$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.u.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.w.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.x.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.x.2 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.y.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.ba.1 $136$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.u.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.w.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.x.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.x.2 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.y.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.ba.1 $152$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.u.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.w.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.x.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.x.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.y.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.ba.1 $168$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.u.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.w.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.x.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.x.2 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.y.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.ba.1 $184$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.u.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.w.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.x.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.x.2 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.y.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.ba.1 $232$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.u.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.w.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.x.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.x.2 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.y.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.ba.1 $248$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.u.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.w.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.x.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.x.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.y.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.ba.1 $264$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.u.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.w.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.x.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.x.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.y.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.ba.1 $280$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.u.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.w.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.x.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.x.2 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.y.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.ba.1 $296$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.u.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.w.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.x.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.x.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.y.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.ba.1 $312$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.u.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.w.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.x.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.x.2 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.y.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.ba.1 $328$ $2$ $2$ $1$ $?$ dimension zero