Invariants
| Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
| Index: | $20$ | $\PSL_2$-index: | $20$ | ||||
| Genus: | $1 = 1 + \frac{ 20 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $10^{2}$ | Cusp orbits | $2$ | ||
| 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: | 10C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.20.1.1 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&12\\3&23\end{bmatrix}$, $\begin{bmatrix}9&12\\8&11\end{bmatrix}$, $\begin{bmatrix}37&20\\23&13\end{bmatrix}$, $\begin{bmatrix}39&7\\23&12\end{bmatrix}$ |
| 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: | $36864$ |
Jacobian
| Conductor: | $2^{6}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.w |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x z - x w + y z - y w $ |
| $=$ | $6 x^{2} + 8 x y + 16 y^{2} - z^{2} + 3 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 16 x^{3} z - 54 x^{2} y^{2} + 17 x^{2} z^{2} + 52 x y^{2} z - 7 x z^{3} - 14 y^{2} z^{2} + z^{4} $ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 20 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 5^3\,\frac{317634048xy^{5}+17556672xy^{3}w^{2}-5594808xyw^{4}+244477440y^{6}+115146304y^{4}w^{2}-255928y^{2}w^{4}-34816z^{6}+186880z^{5}w-1534656z^{4}w^{2}+4368424z^{3}w^{3}-6566455z^{2}w^{4}+3954621zw^{5}-928931w^{6}}{22977000xy^{5}+2315500xy^{3}w^{2}+14250xyw^{4}+17685000y^{6}+5371000y^{4}w^{2}+121750y^{2}w^{4}-2327z^{6}+18112z^{5}w-67714z^{4}w^{2}+160194z^{3}w^{3}-209749z^{2}w^{4}+108617zw^{5}-18512w^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.2.0.a.1 | $8$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.2.0.a.1 | $8$ | $10$ | $10$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 40.40.1.b.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.c.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.e.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.f.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.n.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.o.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.q.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.r.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.60.3.a.1 | $40$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 40.60.3.r.1 | $40$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 40.80.5.a.1 | $40$ | $4$ | $4$ | $5$ | $4$ | $1^{4}$ |
| 120.40.1.b.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.c.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.e.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.f.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.n.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.o.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.q.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.r.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.60.5.bw.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.80.5.a.1 | $120$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 200.100.5.a.1 | $200$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 280.40.1.b.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.c.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.e.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.f.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.n.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.o.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.q.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.r.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.160.11.a.1 | $280$ | $8$ | $8$ | $11$ | $?$ | not computed |