Invariants
Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
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: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.1.71 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&2\\2&3\end{bmatrix}$, $\begin{bmatrix}5&5\\2&3\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: | $C_2.C_4^2$ |
Contains $-I$: | yes |
Quadratic refinements: | 16.96.1-8.bf.1.1, 16.96.1-8.bf.1.2, 16.96.1-8.bf.1.3, 48.96.1-8.bf.1.1, 48.96.1-8.bf.1.2, 48.96.1-8.bf.1.3, 80.96.1-8.bf.1.1, 80.96.1-8.bf.1.2, 80.96.1-8.bf.1.3, 112.96.1-8.bf.1.1, 112.96.1-8.bf.1.2, 112.96.1-8.bf.1.3, 176.96.1-8.bf.1.1, 176.96.1-8.bf.1.2, 176.96.1-8.bf.1.3, 208.96.1-8.bf.1.1, 208.96.1-8.bf.1.2, 208.96.1-8.bf.1.3, 240.96.1-8.bf.1.1, 240.96.1-8.bf.1.2, 240.96.1-8.bf.1.3, 272.96.1-8.bf.1.1, 272.96.1-8.bf.1.2, 272.96.1-8.bf.1.3, 304.96.1-8.bf.1.1, 304.96.1-8.bf.1.2, 304.96.1-8.bf.1.3 |
Cyclic 8-isogeny field degree: | $2$ |
Cyclic 8-torsion field degree: | $8$ |
Full 8-torsion field degree: | $32$ |
Jacobian
Conductor: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + x y + y^{2} + z^{2} - 2 w^{2} $ |
$=$ | $2 x^{2} + x y + 2 y^{2} + z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 12 x^{2} z^{2} + y^{4} + 4 y^{2} z^{2} + 36 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(z^{2}-6w^{2})^{3}(z^{2}-2w^{2})^{3}}{w^{8}(z-2w)^{2}(z+2w)^{2}}$ |
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.s.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
8.24.0.u.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
8.24.0.bg.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
8.24.0.bj.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
8.24.1.p.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
8.24.1.v.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
8.24.1.w.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.bv.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
16.96.5.bx.1 | $16$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
24.144.9.vq.1 | $24$ | $3$ | $3$ | $9$ | $2$ | $1^{8}$ |
24.192.9.ii.1 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
40.240.17.fo.1 | $40$ | $5$ | $5$ | $17$ | $5$ | $1^{14}\cdot2$ |
40.288.17.nj.1 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{14}\cdot2$ |
40.480.33.yc.1 | $40$ | $10$ | $10$ | $33$ | $11$ | $1^{28}\cdot2^{2}$ |
48.96.5.ed.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
48.96.5.ef.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
56.384.25.ii.1 | $56$ | $8$ | $8$ | $25$ | $7$ | $1^{20}\cdot2^{2}$ |
56.1008.73.vq.1 | $56$ | $21$ | $21$ | $73$ | $26$ | $1^{16}\cdot2^{26}\cdot4$ |
56.1344.97.vi.1 | $56$ | $28$ | $28$ | $97$ | $33$ | $1^{36}\cdot2^{28}\cdot4$ |
80.96.5.eh.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.ej.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.ed.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.96.5.ef.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.96.5.ed.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.96.5.ef.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.96.5.eh.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.96.5.ej.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.lf.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.lh.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.96.5.eh.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.96.5.ej.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.96.5.ed.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.96.5.ef.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |