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Properties

Label 8.48.1.l.1
Level $8$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $64$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
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: none

Other labels

Cummins and Pauli (CP) label: 8G1
Rouse and Zureick-Brown (RZB) label: X270
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.41

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&4\\4&3\end{bmatrix}$, $\begin{bmatrix}3&0\\4&7\end{bmatrix}$, $\begin{bmatrix}3&6\\0&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&3\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.96.1-8.l.1.1, 8.96.1-8.l.1.2, 8.96.1-8.l.1.3, 8.96.1-8.l.1.4, 8.96.1-8.l.1.5, 8.96.1-8.l.1.6, 8.96.1-8.l.1.7, 8.96.1-8.l.1.8, 24.96.1-8.l.1.1, 24.96.1-8.l.1.2, 24.96.1-8.l.1.3, 24.96.1-8.l.1.4, 24.96.1-8.l.1.5, 24.96.1-8.l.1.6, 24.96.1-8.l.1.7, 24.96.1-8.l.1.8, 40.96.1-8.l.1.1, 40.96.1-8.l.1.2, 40.96.1-8.l.1.3, 40.96.1-8.l.1.4, 40.96.1-8.l.1.5, 40.96.1-8.l.1.6, 40.96.1-8.l.1.7, 40.96.1-8.l.1.8, 56.96.1-8.l.1.1, 56.96.1-8.l.1.2, 56.96.1-8.l.1.3, 56.96.1-8.l.1.4, 56.96.1-8.l.1.5, 56.96.1-8.l.1.6, 56.96.1-8.l.1.7, 56.96.1-8.l.1.8, 88.96.1-8.l.1.1, 88.96.1-8.l.1.2, 88.96.1-8.l.1.3, 88.96.1-8.l.1.4, 88.96.1-8.l.1.5, 88.96.1-8.l.1.6, 88.96.1-8.l.1.7, 88.96.1-8.l.1.8, 104.96.1-8.l.1.1, 104.96.1-8.l.1.2, 104.96.1-8.l.1.3, 104.96.1-8.l.1.4, 104.96.1-8.l.1.5, 104.96.1-8.l.1.6, 104.96.1-8.l.1.7, 104.96.1-8.l.1.8, 120.96.1-8.l.1.1, 120.96.1-8.l.1.2, 120.96.1-8.l.1.3, 120.96.1-8.l.1.4, 120.96.1-8.l.1.5, 120.96.1-8.l.1.6, 120.96.1-8.l.1.7, 120.96.1-8.l.1.8, 136.96.1-8.l.1.1, 136.96.1-8.l.1.2, 136.96.1-8.l.1.3, 136.96.1-8.l.1.4, 136.96.1-8.l.1.5, 136.96.1-8.l.1.6, 136.96.1-8.l.1.7, 136.96.1-8.l.1.8, 152.96.1-8.l.1.1, 152.96.1-8.l.1.2, 152.96.1-8.l.1.3, 152.96.1-8.l.1.4, 152.96.1-8.l.1.5, 152.96.1-8.l.1.6, 152.96.1-8.l.1.7, 152.96.1-8.l.1.8, 168.96.1-8.l.1.1, 168.96.1-8.l.1.2, 168.96.1-8.l.1.3, 168.96.1-8.l.1.4, 168.96.1-8.l.1.5, 168.96.1-8.l.1.6, 168.96.1-8.l.1.7, 168.96.1-8.l.1.8, 184.96.1-8.l.1.1, 184.96.1-8.l.1.2, 184.96.1-8.l.1.3, 184.96.1-8.l.1.4, 184.96.1-8.l.1.5, 184.96.1-8.l.1.6, 184.96.1-8.l.1.7, 184.96.1-8.l.1.8, 232.96.1-8.l.1.1, 232.96.1-8.l.1.2, 232.96.1-8.l.1.3, 232.96.1-8.l.1.4, 232.96.1-8.l.1.5, 232.96.1-8.l.1.6, 232.96.1-8.l.1.7, 232.96.1-8.l.1.8, 248.96.1-8.l.1.1, 248.96.1-8.l.1.2, 248.96.1-8.l.1.3, 248.96.1-8.l.1.4, 248.96.1-8.l.1.5, 248.96.1-8.l.1.6, 248.96.1-8.l.1.7, 248.96.1-8.l.1.8, 264.96.1-8.l.1.1, 264.96.1-8.l.1.2, 264.96.1-8.l.1.3, 264.96.1-8.l.1.4, 264.96.1-8.l.1.5, 264.96.1-8.l.1.6, 264.96.1-8.l.1.7, 264.96.1-8.l.1.8, 280.96.1-8.l.1.1, 280.96.1-8.l.1.2, 280.96.1-8.l.1.3, 280.96.1-8.l.1.4, 280.96.1-8.l.1.5, 280.96.1-8.l.1.6, 280.96.1-8.l.1.7, 280.96.1-8.l.1.8, 296.96.1-8.l.1.1, 296.96.1-8.l.1.2, 296.96.1-8.l.1.3, 296.96.1-8.l.1.4, 296.96.1-8.l.1.5, 296.96.1-8.l.1.6, 296.96.1-8.l.1.7, 296.96.1-8.l.1.8, 312.96.1-8.l.1.1, 312.96.1-8.l.1.2, 312.96.1-8.l.1.3, 312.96.1-8.l.1.4, 312.96.1-8.l.1.5, 312.96.1-8.l.1.6, 312.96.1-8.l.1.7, 312.96.1-8.l.1.8, 328.96.1-8.l.1.1, 328.96.1-8.l.1.2, 328.96.1-8.l.1.3, 328.96.1-8.l.1.4, 328.96.1-8.l.1.5, 328.96.1-8.l.1.6, 328.96.1-8.l.1.7, 328.96.1-8.l.1.8
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $8$
Full 8-torsion field degree: $32$

Jacobian

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

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ x^{2} + x y + z^{2} $
$=$ $x^{2} - x y - 4 y^{2} + z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 2 x^{2} y^{2} + 3 x^{2} z^{2} + 2 z^{4} $
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{4032y^{2}z^{10}+6048y^{2}z^{8}w^{2}+288y^{2}z^{6}w^{4}-144y^{2}z^{4}w^{6}-756y^{2}z^{2}w^{8}-126y^{2}w^{10}+2048z^{12}+6144z^{10}w^{2}+4080z^{8}w^{4}+512z^{6}w^{6}+192z^{4}w^{8}-120z^{2}w^{10}-31w^{12}}{w^{4}z^{4}(4y^{2}z^{2}-2y^{2}w^{2}-w^{4})}$

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 Kernel decomposition
8.24.0.d.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.d.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.1.d.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
8.96.1.a.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.a.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.i.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.i.2 $8$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.ba.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.ba.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bi.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bi.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.dx.1 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
24.192.9.cg.1 $24$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
40.96.1.ba.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.ba.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bi.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bi.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.bj.1 $40$ $5$ $5$ $17$ $6$ $1^{6}\cdot2^{5}$
40.288.17.cp.1 $40$ $6$ $6$ $17$ $0$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.fz.1 $40$ $10$ $10$ $33$ $8$ $1^{12}\cdot2^{6}\cdot4^{2}$
56.96.1.ba.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.ba.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bi.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bi.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.cg.1 $56$ $8$ $8$ $25$ $3$ $1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.dx.1 $56$ $21$ $21$ $73$ $9$ $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.dx.1 $56$ $28$ $28$ $97$ $12$ $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
88.96.1.ba.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.ba.2 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.bi.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.bi.2 $88$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.ba.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.ba.2 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.bi.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.bi.2 $104$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ds.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ds.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ei.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ei.2 $120$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.ba.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.ba.2 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.bi.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.bi.2 $136$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.ba.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.ba.2 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.bi.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.bi.2 $152$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ds.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ds.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ei.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ei.2 $168$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.ba.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.ba.2 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.bi.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.bi.2 $184$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.ba.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.ba.2 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.bi.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.bi.2 $232$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.ba.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.ba.2 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.bi.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.bi.2 $248$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ds.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ds.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ei.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ei.2 $264$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ds.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ds.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ei.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ei.2 $280$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.ba.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.ba.2 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.bi.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.bi.2 $296$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ds.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ds.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ei.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ei.2 $312$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.ba.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.ba.2 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.bi.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.bi.2 $328$ $2$ $2$ $1$ $?$ dimension zero