Invariants
Level: | $40$ | $\SL_2$-level: | $10$ | Newform level: | $1600$ | ||
Index: | $40$ | $\PSL_2$-index: | $40$ | ||||
Genus: | $1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $10^{4}$ | Cusp orbits | $4$ | ||
Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 10H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.40.1.2 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}6&3\\9&11\end{bmatrix}$, $\begin{bmatrix}16&11\\27&29\end{bmatrix}$, $\begin{bmatrix}22&31\\13&12\end{bmatrix}$ |
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: | $18432$ |
Jacobian
Conductor: | $2^{6}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 1600.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} + y^{2} + z^{2} - z w + w^{2} $ |
$=$ | $x^{2} - y^{2} + 2 y z + 2 y w - z^{2} - z w - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 2 x^{3} z + 8 x^{2} y^{2} + 4 x^{2} z^{2} + 2 x y^{2} z - 3 x z^{3} + 36 y^{4} + 2 y^{2} z^{2} + 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -3^2\cdot5^2\,\frac{8144yz^{9}-156980yz^{8}w+476032yz^{7}w^{2}-661896yz^{6}w^{3}+323372yz^{5}w^{4}+323372yz^{4}w^{5}-661896yz^{3}w^{6}+476032yz^{2}w^{7}-156980yzw^{8}+8144yw^{9}+17004z^{10}+25084z^{9}w-231923z^{8}w^{2}+578086z^{7}w^{3}-891049z^{6}w^{4}+1005404z^{5}w^{5}-891049z^{4}w^{6}+578086z^{3}w^{7}-231923z^{2}w^{8}+25084zw^{9}+17004w^{10}}{95yz^{9}+285yz^{8}w+375yz^{7}w^{2}+465yz^{6}w^{3}+255yz^{5}w^{4}+255yz^{4}w^{5}+465yz^{3}w^{6}+375yz^{2}w^{7}+285yzw^{8}+95yw^{9}-24z^{10}-155z^{9}w-70z^{8}w^{2}-5z^{7}w^{3}+80z^{6}w^{4}+373z^{5}w^{5}+80z^{4}w^{6}-5z^{3}w^{7}-70z^{2}w^{8}-155zw^{9}-24w^{10}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
5.20.0.b.1 | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
8.2.0.b.1 | $8$ | $20$ | $20$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
5.20.0.b.1 | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.10.0.b.1 | $40$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
40.20.0.e.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.20.1.b.1 | $40$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
40.120.5.q.1 | $40$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
40.120.5.cp.1 | $40$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
40.160.9.t.1 | $40$ | $4$ | $4$ | $9$ | $4$ | $1^{8}$ |
120.120.9.fv.1 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.160.9.bf.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
200.200.9.l.1 | $200$ | $5$ | $5$ | $9$ | $?$ | not computed |
280.320.21.t.1 | $280$ | $8$ | $8$ | $21$ | $?$ | not computed |