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 | $2^{2}\cdot4$ | ||
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, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.1.93 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\4&3\end{bmatrix}$, $\begin{bmatrix}7&7\\2&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: | $D_4:C_4$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 8-isogeny field degree: | $4$ |
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 + x z + y^{2} + z^{2} $ |
$=$ | $2 x^{2} + 2 x y - 2 x z + y^{2} - 2 y z + z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 13 x^{4} - 28 x^{3} y + 46 x^{2} y^{2} - 4 x^{2} z^{2} - 28 x y^{3} + 24 x y z^{2} + 13 y^{4} + \cdots + 4 z^{4} $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^7\,\frac{179317670400xz^{11}+3146612790528xz^{9}w^{2}+15632192399616xz^{7}w^{4}+71530599317632xz^{5}w^{6}+31600182934944xz^{3}w^{8}-319593934192xzw^{10}-591032855520y^{2}z^{10}-6497213259888y^{2}z^{8}w^{2}-26881436355696y^{2}z^{6}w^{4}-18659292102360y^{2}z^{4}w^{6}+6652192149018y^{2}z^{2}w^{8}+423001119645y^{2}w^{10}+614230914240yz^{11}+6559817955552yz^{9}w^{2}+28092810436704yz^{7}w^{4}+36756972165680yz^{5}w^{6}+15973477784732yz^{3}w^{8}+2756643914726yzw^{10}-352736780832z^{12}-3913540013232z^{10}w^{2}-14139702618912z^{8}w^{4}+2251170228424z^{6}w^{6}+46458726794974z^{4}w^{8}+7244688466041z^{2}w^{10}+445180307w^{12}}{13282790400xz^{11}-111547000320xz^{9}w^{2}-202665868288xz^{7}w^{4}-48545582592xz^{5}w^{6}+21138022464xz^{3}w^{8}+4758948064xzw^{10}-43780211520y^{2}z^{10}-136768856432y^{2}z^{8}w^{2}+95586221536y^{2}z^{6}w^{4}+115703177096y^{2}z^{4}w^{6}-7381647572y^{2}z^{2}w^{8}-373892051y^{2}w^{10}+45498586240yz^{11}-84151009376yz^{9}w^{2}-314019045184yz^{7}w^{4}+39445933072yz^{5}w^{6}+59281334632yz^{3}w^{8}-5433844494yzw^{10}-26128650432z^{12}+13358142736z^{10}w^{2}+1353521728z^{8}w^{4}-136540274104z^{6}w^{6}+9361654276z^{4}w^{8}+16054623637z^{2}w^{10}-403224198w^{12}}$ |
Modular covers
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.bk.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
8.24.0.bl.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
8.24.1.e.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 |
---|---|---|---|---|---|---|
16.96.5.bc.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
16.96.5.bc.3 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
16.96.5.bd.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
16.96.5.bd.3 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
24.144.9.nd.2 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
24.192.9.gj.1 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
40.240.17.df.2 | $40$ | $5$ | $5$ | $17$ | $3$ | $1^{6}\cdot2^{5}$ |
40.288.17.ic.1 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{6}\cdot2\cdot4^{2}$ |
40.480.33.pp.1 | $40$ | $10$ | $10$ | $33$ | $4$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
48.96.5.cr.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.cr.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.ct.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.ct.3 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
56.384.25.gj.1 | $56$ | $8$ | $8$ | $25$ | $1$ | $1^{8}\cdot2^{4}\cdot4^{2}$ |
56.1008.73.nd.1 | $56$ | $21$ | $21$ | $73$ | $5$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
56.1344.97.nd.2 | $56$ | $28$ | $28$ | $97$ | $6$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
80.96.5.cv.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.cv.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.cx.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.cx.3 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.cr.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.cr.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.ct.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.ct.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.96.5.cr.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.96.5.cr.3 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.96.5.ct.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.96.5.ct.3 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.96.5.cv.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.96.5.cv.3 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.96.5.cx.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.96.5.cx.3 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.hp.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.hp.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.ht.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.ht.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.96.5.cv.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.96.5.cv.3 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.96.5.cx.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.96.5.cx.3 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.96.5.cr.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.96.5.cr.3 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.96.5.ct.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.96.5.ct.3 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |