Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 20H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.1.87 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&25\\28&39\end{bmatrix}$, $\begin{bmatrix}11&21\\38&9\end{bmatrix}$, $\begin{bmatrix}15&22\\14&13\end{bmatrix}$, $\begin{bmatrix}21&37\\4&39\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $4$ |
Cyclic 40-torsion field degree: | $64$ |
Full 40-torsion field degree: | $10240$ |
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 $ | $=$ | $ 5 x y + w^{2} $ |
$=$ | $x^{2} - 2 x y + 5 y^{2} + 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 10 x^{2} y^{2} + 2 x^{2} z^{2} + 5 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}z$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{15257812500y^{2}z^{16}-21360937500y^{2}z^{14}w^{2}+9452812500y^{2}z^{12}w^{4}-173250000y^{2}z^{10}w^{6}-1075500000y^{2}z^{8}w^{8}+320670000y^{2}z^{6}w^{10}-34218000y^{2}z^{4}w^{12}+896400y^{2}z^{2}w^{14}+23760y^{2}w^{16}+6103515625z^{18}-7324218750z^{16}w^{2}+1770000000z^{14}w^{4}+1171906250z^{12}w^{6}-738075000z^{10}w^{8}+125475000z^{8}w^{10}+3557500z^{6}w^{12}-3000600z^{4}w^{14}+239760z^{2}w^{16}-4104w^{18}}{w^{4}(5z^{2}-2w^{2})(125000y^{2}z^{10}-125000y^{2}z^{8}w^{2}+12500y^{2}z^{6}w^{4}+22500y^{2}z^{4}w^{6}-7500y^{2}z^{2}w^{8}+440y^{2}w^{10}+2500z^{8}w^{4}-3000z^{6}w^{6}+975z^{4}w^{8}+80z^{2}w^{10}-76w^{12})}$ |
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 |
---|---|---|---|---|---|---|
20.36.0.c.2 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.0.a.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.1.g.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 |
---|---|---|---|---|---|---|
40.144.5.o.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.bz.2 | $40$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
40.144.5.cu.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.cy.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.ih.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.im.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.iv.2 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
40.144.5.jh.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.360.13.cb.1 | $40$ | $5$ | $5$ | $13$ | $3$ | $1^{6}\cdot2^{3}$ |
120.144.5.che.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.chi.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cig.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.cik.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egt.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.egw.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ehv.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ehy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.uk.1 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.ieo.1 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.bm.1 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.bfx.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bfy.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgl.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bgm.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bon.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.boo.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bpb.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bpc.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |