Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
| Index: | $160$ | $\PSL_2$-index: | $160$ | ||||
| Genus: | $9 = 1 + \frac{ 160 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $20^{8}$ | Cusp orbits | $8$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.160.9.6 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}16&39\\33&19\end{bmatrix}$, $\begin{bmatrix}27&28\\7&9\end{bmatrix}$, $\begin{bmatrix}34&17\\3&11\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $72$ |
| Cyclic 40-torsion field degree: | $1152$ |
| Full 40-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{48}\cdot5^{18}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{9}$ |
| Newforms: | 400.2.a.a, 400.2.a.e, 400.2.a.f, 1600.2.a.a, 1600.2.a.c, 1600.2.a.f, 1600.2.a.q, 1600.2.a.v, 1600.2.a.w |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x v - y v - z t - t r + v r $ |
| $=$ | $t^{2} + t u - u v + v^{2}$ | |
| $=$ | $x t + y t + z v - w t - w v$ | |
| $=$ | $x u - y t - y v - w u + w v + v r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 169 x^{16} + 416 x^{15} z - 1372 x^{14} y^{2} + 464 x^{14} z^{2} + 776 x^{13} y^{2} z + 2752 x^{13} z^{3} + \cdots + 169 z^{16} $ |
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 canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 40.80.5.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w+r$ |
| $\displaystyle W$ | $=$ | $\displaystyle -t-v$ |
| $\displaystyle T$ | $=$ | $\displaystyle -t-u+v$ |
Equation of the image curve:
| $0$ | $=$ | $ XW+YW-2ZW-2XT-YT-ZT $ |
| $=$ | $ 10Y^{2}+2Z^{2}-3W^{2}-2WT-2T^{2} $ | |
| $=$ | $ 18X^{2}-28XY+2Y^{2}+4XZ-12YZ-2Z^{2}-5T^{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 |
|---|---|---|---|---|---|---|
| 20.80.3.b.1 | $20$ | $2$ | $2$ | $3$ | $2$ | $1^{6}$ |
| 40.40.1.o.1 | $40$ | $4$ | $4$ | $1$ | $0$ | $1^{8}$ |
| 40.80.5.a.1 | $40$ | $2$ | $2$ | $5$ | $4$ | $1^{4}$ |
| 40.80.5.g.1 | $40$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.29.kk.1 | $40$ | $3$ | $3$ | $29$ | $14$ | $1^{20}$ |
| 40.480.29.bbe.1 | $40$ | $3$ | $3$ | $29$ | $13$ | $1^{20}$ |
| 40.640.45.t.1 | $40$ | $4$ | $4$ | $45$ | $22$ | $1^{26}\cdot2^{5}$ |
| 80.320.21.l.1 | $80$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 80.320.21.n.1 | $80$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 80.320.21.z.1 | $80$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 80.320.21.bd.1 | $80$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.320.21.bb.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.320.21.bd.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.320.21.cv.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.320.21.cz.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |