Invariants
Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $1600$ | ||
Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 20J3 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.221 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&24\\2&17\end{bmatrix}$, $\begin{bmatrix}7&4\\10&1\end{bmatrix}$, $\begin{bmatrix}15&6\\26&5\end{bmatrix}$, $\begin{bmatrix}23&34\\16&1\end{bmatrix}$, $\begin{bmatrix}29&4\\30&33\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.72.3.d.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $4$ |
Cyclic 40-torsion field degree: | $32$ |
Full 40-torsion field degree: | $5120$ |
Jacobian
Conductor: | $2^{14}\cdot5^{5}$ |
Simple: | no |
Squarefree: | yes |
Decomposition: | $1^{3}$ |
Newforms: | 20.2.a.a, 1600.2.a.k, 1600.2.a.w |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
$ 0 $ | $=$ | $ x y w - x y t + y z t $ |
$=$ | $x w t - x t^{2} + z t^{2}$ | |
$=$ | $x w^{2} - x w t + z w t$ | |
$=$ | $x^{2} w - x^{2} t + x z t$ | |
$=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} y + x^{2} y^{2} z - 40 x^{2} y z^{2} + 500 x^{2} z^{3} + 100 y z^{4} $ |
Weierstrass model Weierstrass model
$ y^{2} + x^{4} y $ | $=$ | $ -20x^{6} - 50x^{4} - 2000x^{2} + 2500 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Embedded model |
---|
$(0:0:0:0:1)$, $(1:0:1:0:0)$, $(0:1:0:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{70312500000xz^{10}+11256093750000xz^{8}t^{2}-100597367750000xz^{6}t^{4}+46719924455500xz^{4}t^{6}+5757951566760xz^{2}t^{8}-88848484174xt^{10}+31250000000y^{11}+16406250000y^{9}w^{2}-457031250000y^{9}wt+468437500000y^{9}t^{2}+2172031250000y^{7}w^{2}t^{2}-2276355625000y^{7}wt^{3}+7971556625000y^{7}t^{4}+977086500000y^{5}w^{2}t^{4}-2725818760000y^{5}wt^{5}+646715893000y^{5}t^{6}+282203506750y^{3}w^{2}t^{6}-79070898010y^{3}wt^{7}+69027617100y^{3}t^{8}-15051009495yw^{2}t^{8}+268325929195ywt^{9}-214325485772yt^{10}+125000000000z^{11}-485156250000z^{9}w^{2}-2311718750000z^{9}wt-11053750000000z^{9}t^{2}+19197328125000z^{7}w^{2}t^{2}+30769799375000z^{7}wt^{3}+101606649000000z^{7}t^{4}-71636957625000z^{5}w^{2}t^{4}-7134208715000z^{5}wt^{5}-53206603168000z^{5}t^{6}+3046013988000z^{3}w^{2}t^{6}-7763870950290z^{3}wt^{7}-4805034948960z^{3}t^{8}+1153429858657zw^{2}t^{8}-723540728847zwt^{9}+303298969946zt^{10}}{t(13000000000xz^{8}t-1884000000xz^{6}t^{3}-256880000xz^{4}t^{5}+768480xz^{2}t^{7}+48185xt^{9}-625000000y^{9}w+656250000y^{9}t-396875000y^{7}w^{2}t+422500000y^{7}wt^{2}+208500000y^{7}t^{3}+30775000y^{5}w^{2}t^{3}+44785000y^{5}wt^{4}-13790000y^{5}t^{5}+9222800y^{3}w^{2}t^{5}-2612450y^{3}wt^{6}-1636950y^{3}t^{7}-140228yw^{2}t^{7}+376656ywt^{8}-173973yt^{9}-2500000000z^{9}w-13000000000z^{9}t+12350000000z^{7}w^{2}t-960000000z^{7}wt^{2}+2934000000z^{7}t^{3}+1811600000z^{5}w^{2}t^{3}-622160000z^{5}wt^{4}+472480000z^{5}t^{5}+15231200z^{3}w^{2}t^{5}-19812750z^{3}wt^{6}+7527520z^{3}t^{7}+17277zw^{2}t^{7}-280855zwt^{8}+125788zt^{9})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.d.1 :
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle t$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{10}w$ |
Equation of the image curve:
$0$ | $=$ | $ X^{4}Y+X^{2}Y^{2}Z-40X^{2}YZ^{2}+500X^{2}Z^{3}+100YZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.d.1 :
$\displaystyle X$ | $=$ | $\displaystyle -w$ |
$\displaystyle Y$ | $=$ | $\displaystyle -50y^{4}+20y^{2}w^{2}-10y^{2}wt-w^{4}$ |
$\displaystyle Z$ | $=$ | $\displaystyle y$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
20.72.1-10.a.1.1 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
40.24.0-40.b.1.4 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
40.288.5-40.p.1.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
40.288.5-40.p.2.5 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
40.288.5-40.r.1.3 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
40.288.5-40.r.2.3 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
40.288.5-40.v.1.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
40.288.5-40.v.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
40.288.5-40.x.1.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
40.288.5-40.x.2.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $2$ |
40.288.7-40.f.1.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
40.288.7-40.f.1.2 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
40.288.7-40.g.1.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
40.288.7-40.g.1.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
40.288.7-40.h.1.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
40.288.7-40.h.1.4 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
40.288.7-40.i.1.5 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
40.288.7-40.i.1.14 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
40.288.7-40.bi.1.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
40.288.7-40.bi.1.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
40.288.7-40.bi.2.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
40.288.7-40.bi.2.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
40.288.7-40.bj.1.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
40.288.7-40.bj.1.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
40.288.7-40.bj.2.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
40.288.7-40.bj.2.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
40.720.19-40.p.1.1 | $40$ | $5$ | $5$ | $19$ | $5$ | $1^{16}$ |
120.288.5-120.et.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.288.5-120.et.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.288.5-120.ev.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.288.5-120.ev.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.288.5-120.ez.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.288.5-120.ez.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.288.5-120.fb.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.288.5-120.fb.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.288.7-120.eo.1.25 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.eo.1.29 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.eq.1.17 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.eq.1.23 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.eu.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.eu.1.23 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.ew.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.ew.1.25 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bdl.1.4 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bdl.1.11 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bdl.2.4 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bdl.2.7 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bdm.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bdm.1.14 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bdm.2.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bdm.2.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.432.15-120.f.1.4 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
280.288.5-280.bn.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.288.5-280.bn.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.288.5-280.bp.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.288.5-280.bp.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.288.5-280.bt.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.288.5-280.bt.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.288.5-280.bv.1.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.288.5-280.bv.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
280.288.7-280.x.1.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.x.1.6 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.y.1.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.y.1.6 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.ba.1.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.ba.1.14 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.bb.1.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.bb.1.14 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.by.1.3 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.by.1.8 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.by.2.3 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.by.2.8 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.bz.1.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.bz.1.8 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.bz.2.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.288.7-280.bz.2.8 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |