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 | $2^{4}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ 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: | 20H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.72.1.64 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}25&7\\22&25\end{bmatrix}$, $\begin{bmatrix}33&24\\22&35\end{bmatrix}$, $\begin{bmatrix}37&26\\10&13\end{bmatrix}$, $\begin{bmatrix}39&35\\20&19\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $4$ |
Cyclic 40-torsion field degree: | $32$ |
Full 40-torsion field degree: | $10240$ |
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^{2} - x z - z^{2} - z w $ |
$=$ | $x^{2} - 2 x z - 2 x w + 10 y^{2} + 2 z^{2} + 2 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{3} z + 4 x^{2} z^{2} + 10 y^{2} z^{2} + 4 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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{1913145xz^{17}+11629804xz^{16}w-9291456xz^{15}w^{2}-217591200xz^{14}w^{3}-445609200xz^{13}w^{4}+712615872xz^{12}w^{5}+4269447936xz^{11}w^{6}+5861175296xz^{10}w^{7}-3159133440xz^{9}w^{8}-23312010240xz^{8}w^{9}-40752766976xz^{7}w^{10}-41688416256xz^{6}w^{11}-28181078016xz^{5}w^{12}-12874383360xz^{4}w^{13}-3883008000xz^{3}w^{14}-720371712xz^{2}w^{15}-70189056xzw^{16}-2359296xw^{17}-1911097z^{18}-12259327z^{17}w+5568402z^{16}w^{2}+221493712z^{15}w^{3}+517104360z^{14}w^{4}-579707184z^{13}w^{5}-4533137568z^{12}w^{6}-7220678912z^{11}w^{7}+1492405248z^{10}w^{8}+24675203840z^{9}w^{9}+48164158976z^{8}w^{10}+53715468288z^{7}w^{11}+39728975872z^{6}w^{12}+20156669952z^{5}w^{13}+6938050560z^{4}w^{14}+1542324224z^{3}w^{15}+198770688z^{2}w^{16}+11599872zw^{17}+131072w^{18}}{z^{10}(z+w)^{4}(81xz^{3}-108xz^{2}w-248xzw^{2}-64xw^{3}-81z^{4}+81z^{3}w+290z^{2}w^{2}+136zw^{3}+8w^{4})}$ |
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 |
---|---|---|---|---|---|---|
20.36.0.b.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.0.a.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.36.1.h.1 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.r.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.bx.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.dh.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.dm.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.ht.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.if.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
40.144.5.jj.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.144.5.jo.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
40.360.13.by.1 | $40$ | $5$ | $5$ | $13$ | $2$ | $1^{6}\cdot2^{3}$ |
120.144.5.chs.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.chw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ciu.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ciy.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ehh.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.ehk.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eij.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.eim.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.216.13.uq.1 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.288.13.ieu.1 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
200.360.13.bs.1 | $200$ | $5$ | $5$ | $13$ | $?$ | not computed |
280.144.5.bgz.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bha.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bhg.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bhh.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bpp.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bpq.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bpw.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.144.5.bpx.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |