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: | $2$ | ||||||
| $\Q$-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: | 40.144.3.237 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&6\\36&15\end{bmatrix}$, $\begin{bmatrix}11&28\\2&27\end{bmatrix}$, $\begin{bmatrix}21&38\\28&11\end{bmatrix}$, $\begin{bmatrix}33&8\\30&11\end{bmatrix}$, $\begin{bmatrix}35&2\\2&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.3.c.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| 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.c, 1600.2.a.o |
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.c.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.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 50y^{4}+20y^{2}w^{2}-10y^{2}wt$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
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.1.6 | $20$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 40.24.0-40.a.1.7 | $40$ | $6$ | $6$ | $0$ | $0$ | 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.m.1.18 | $40$ | $2$ | $2$ | $5$ | $2$ | $2$ |
| 40.288.5-40.m.2.17 | $40$ | $2$ | $2$ | $5$ | $2$ | $2$ |
| 40.288.5-40.o.1.8 | $40$ | $2$ | $2$ | $5$ | $2$ | $2$ |
| 40.288.5-40.o.2.7 | $40$ | $2$ | $2$ | $5$ | $2$ | $2$ |
| 40.288.5-40.s.1.8 | $40$ | $2$ | $2$ | $5$ | $2$ | $2$ |
| 40.288.5-40.s.2.5 | $40$ | $2$ | $2$ | $5$ | $2$ | $2$ |
| 40.288.5-40.u.1.8 | $40$ | $2$ | $2$ | $5$ | $2$ | $2$ |
| 40.288.5-40.u.2.7 | $40$ | $2$ | $2$ | $5$ | $2$ | $2$ |
| 40.288.7-40.d.1.2 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 40.288.7-40.e.1.6 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 40.288.7-40.e.1.10 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 40.288.7-40.h.1.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.288.7-40.h.1.12 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.288.7-40.j.1.6 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 40.288.7-40.j.1.9 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 40.288.7-40.be.1.8 | $40$ | $2$ | $2$ | $7$ | $2$ | $2^{2}$ |
| 40.288.7-40.be.1.14 | $40$ | $2$ | $2$ | $7$ | $2$ | $2^{2}$ |
| 40.288.7-40.be.2.10 | $40$ | $2$ | $2$ | $7$ | $2$ | $2^{2}$ |
| 40.288.7-40.be.2.16 | $40$ | $2$ | $2$ | $7$ | $2$ | $2^{2}$ |
| 40.288.7-40.bg.1.8 | $40$ | $2$ | $2$ | $7$ | $2$ | $2^{2}$ |
| 40.288.7-40.bg.1.13 | $40$ | $2$ | $2$ | $7$ | $2$ | $2^{2}$ |
| 40.288.7-40.bg.2.9 | $40$ | $2$ | $2$ | $7$ | $2$ | $2^{2}$ |
| 40.288.7-40.bg.2.16 | $40$ | $2$ | $2$ | $7$ | $2$ | $2^{2}$ |
| 40.720.19-40.m.1.7 | $40$ | $5$ | $5$ | $19$ | $7$ | $1^{16}$ |
| 120.288.5-120.eq.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eq.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.es.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.es.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ew.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ew.2.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ey.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ey.2.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-120.el.1.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.el.1.30 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.en.1.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.en.1.30 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.er.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.er.1.24 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.et.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.et.1.29 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdh.1.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdh.1.30 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdh.2.22 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdh.2.28 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdj.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdj.1.28 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdj.2.20 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdj.2.29 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.432.15-120.e.1.54 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-280.bk.1.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bk.2.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bm.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bm.2.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bq.1.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bq.2.16 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bs.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bs.2.16 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-280.r.1.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.r.1.30 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.s.1.9 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.s.1.30 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.u.1.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.u.1.30 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.v.1.14 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.v.1.17 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bu.1.16 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bu.1.26 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bu.2.20 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bu.2.30 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bw.1.15 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bw.1.26 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bw.2.20 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.bw.2.29 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |