Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $800$ | ||
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 | $4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ 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: | 8G1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.1.216 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&34\\23&11\end{bmatrix}$, $\begin{bmatrix}5&34\\31&21\end{bmatrix}$, $\begin{bmatrix}7&6\\33&5\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $384$ |
Full 40-torsion field degree: | $15360$ |
Jacobian
Conductor: | $2^{5}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 800.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + x w - 2 y^{2} - z^{2} - w^{2} $ |
$=$ | $3 x^{2} - 2 x w - 4 y z + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 24 x^{3} y + 52 x^{2} y^{2} + 20 x^{2} z^{2} - 48 x y^{3} - 60 x y z^{2} + 36 y^{4} + \cdots + 25 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3^8}\cdot\frac{162035040xyz^{9}w+378374400xyz^{7}w^{3}-7336224000xyz^{5}w^{5}-662400000xyz^{3}w^{7}+1848000000xyzw^{9}-33169000xz^{10}w-2015298000xz^{8}w^{3}-222600000xz^{6}w^{5}+10554960000xz^{4}w^{7}+414000000xz^{2}w^{9}-308000000xw^{11}-13448480yz^{11}-366410640yz^{9}w^{2}+5293286400yz^{7}w^{4}+6795600000yz^{5}w^{6}-9316800000yz^{3}w^{8}-42400000yzw^{10}+656529z^{12}+505604200z^{10}w^{2}+1033888500z^{8}w^{4}-5803800000z^{6}w^{6}-6977730000z^{4}w^{8}+2782800000z^{2}w^{10}+23000000w^{12}}{z^{8}(120xyzw-50xz^{2}w-100xw^{3}-40yz^{3}-20yzw^{2}+3z^{4}+50z^{2}w^{2}+25w^{4})}$ |
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.bk.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.0.el.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.24.1.ee.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 |
---|---|---|---|---|---|---|
40.240.17.yo.1 | $40$ | $5$ | $5$ | $17$ | $6$ | $1^{6}\cdot2^{5}$ |
40.288.17.cgn.2 | $40$ | $6$ | $6$ | $17$ | $3$ | $1^{6}\cdot2\cdot4^{2}$ |
40.480.33.dxs.2 | $40$ | $10$ | $10$ | $33$ | $7$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
80.96.5.ht.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.hv.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.jp.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.jr.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.sv.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.sx.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.ur.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.96.5.ut.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.9.beyo.2 | $120$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.192.9.doz.1 | $120$ | $4$ | $4$ | $9$ | $?$ | not computed |
240.96.5.cfv.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cfx.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cgt.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cgv.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.drf.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.drh.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.dsd.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.dsf.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |