Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $32$ | $\PSL_2$-index: | $32$ | ||||
| Genus: | $1 = 1 + \frac{ 32 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $8^{4}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ 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: | 8E1 |
| Rouse and Zureick-Brown (RZB) label: | X179 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.32.1.3 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\7&7\end{bmatrix}$, $\begin{bmatrix}1&7\\1&2\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_6:C_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 8-isogeny field degree: | $6$ |
| Cyclic 8-torsion field degree: | $24$ |
| Full 8-torsion field degree: | $48$ |
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} + 2 x y - 2 z w $ |
| $=$ | $3 x^{2} - 2 x y - 4 x z + 4 x w + 2 y^{2} - z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 8 x^{3} y + 8 x^{3} z - 2 x^{2} y^{2} - 8 x^{2} y z - 2 x^{2} z^{2} + 4 y^{2} z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 32 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\cdot3^3\,\frac{(z-w)(10870544xz^{6}+14489216xz^{5}w+6584944xz^{4}w^{2}-57917440xz^{3}w^{3}+6584944xz^{2}w^{4}+14489216xzw^{5}+10870544xw^{6}-5101676y^{2}z^{5}-3088292y^{2}z^{4}w-7777016y^{2}z^{3}w^{2}+7777016y^{2}z^{2}w^{3}+3088292y^{2}zw^{4}+5101676y^{2}w^{5}+472392yz^{6}-8262800yz^{5}w+1455800yz^{4}w^{2}+6697248yz^{3}w^{3}+1455800yz^{2}w^{4}-8262800yzw^{5}+472392yw^{6}+2216227z^{7}+10247811z^{6}w-2175737z^{5}w^{2}-4984729z^{4}w^{3}+4984729z^{3}w^{4}+2175737z^{2}w^{5}-10247811zw^{6}-2216227w^{7})}{457924xz^{7}-599028xz^{6}w+482356xz^{5}w^{2}-456004xz^{4}w^{3}+456004xz^{3}w^{4}-482356xz^{2}w^{5}+599028xzw^{6}-457924xw^{7}-190258y^{2}z^{6}+189096y^{2}z^{5}w-155058y^{2}z^{4}w^{2}+154976y^{2}z^{3}w^{3}-155058y^{2}z^{2}w^{4}+189096y^{2}zw^{5}-190258y^{2}w^{6}-154816yz^{6}w+145088yz^{5}w^{2}-127744yz^{4}w^{3}+127744yz^{3}w^{4}-145088yz^{2}w^{5}+154816yzw^{6}+95129z^{8}+95710z^{7}w-158206z^{6}w^{2}+106862z^{5}w^{3}-121526z^{4}w^{4}+106862z^{3}w^{5}-158206z^{2}w^{6}+95710zw^{7}+95129w^{8}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.8.0.a.1 | $8$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(8)$ | $8$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 8.96.3.k.1 | $8$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 16.64.1.a.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.64.1.b.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.128.7.b.1 | $16$ | $4$ | $4$ | $7$ | $3$ | $1^{4}\cdot2$ |
| 24.96.7.b.1 | $24$ | $3$ | $3$ | $7$ | $3$ | $1^{6}$ |
| 24.128.7.b.1 | $24$ | $4$ | $4$ | $7$ | $1$ | $1^{6}$ |
| 40.160.11.b.1 | $40$ | $5$ | $5$ | $11$ | $6$ | $1^{8}\cdot2$ |
| 40.192.13.j.1 | $40$ | $6$ | $6$ | $13$ | $2$ | $1^{10}\cdot2$ |
| 40.320.23.b.1 | $40$ | $10$ | $10$ | $23$ | $13$ | $1^{18}\cdot2^{2}$ |
| 48.64.1.a.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.64.1.b.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.256.17.b.1 | $56$ | $8$ | $8$ | $17$ | $5$ | $1^{12}\cdot2^{2}$ |
| 56.672.51.b.1 | $56$ | $21$ | $21$ | $51$ | $31$ | $1^{10}\cdot2^{18}\cdot4$ |
| 56.896.67.b.1 | $56$ | $28$ | $28$ | $67$ | $36$ | $1^{22}\cdot2^{20}\cdot4$ |
| 80.64.1.a.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.64.1.b.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.64.1.a.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.64.1.b.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.64.1.a.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.64.1.b.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.64.1.a.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.64.1.b.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.64.1.a.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.64.1.b.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.64.1.a.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.64.1.b.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.64.1.a.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.64.1.b.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |