Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $40$ | ||
| 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 $6$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $1^{6}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.1009 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&38\\20&9\end{bmatrix}$, $\begin{bmatrix}5&12\\2&15\end{bmatrix}$, $\begin{bmatrix}9&20\\22&7\end{bmatrix}$, $\begin{bmatrix}19&22\\8&13\end{bmatrix}$, $\begin{bmatrix}27&2\\30&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.72.3.e.2 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: | $2^{8}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 20.2.a.a, 40.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ - x z t + y z w $ |
| $=$ | $ - x y t + y^{2} w$ | |
| $=$ | $ - x t^{2} + y w t$ | |
| $=$ | $ - x w t + y w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y + x^{4} z - x^{2} y^{2} z - 6 x^{2} y z^{2} - 6 x^{2} z^{3} + 5 y z^{4} + 5 z^{5} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ -2x^{6} + 5x^{4} - 10x^{2} + 6 $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:1:0)$, $(0:0:1:0:0)$, $(0:-1/5:-1/5:0:1)$, $(-1/2:0:1:0:0)$, $(0:1/5:1/5:0:1)$, $(1/2:-1/2:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2}\cdot\frac{23999921875000xz^{10}+5671906250000xz^{8}t^{2}+5842667000000xz^{6}t^{4}-58344341208750xz^{4}t^{6}-77036349944575xz^{2}t^{8}-14738786322236xw^{2}t^{8}-29318720156472xwt^{9}-6909924601118xt^{10}+47992421875000yz^{10}+11274718750000yz^{8}t^{2}+7780530750000yz^{6}t^{4}-58267189488750yz^{4}t^{6}-68252899650375yz^{2}t^{8}+22467058307354yt^{10}+12000000000000z^{11}-656250000z^{9}w^{2}+4800218750000z^{9}wt+12402375000000z^{9}t^{2}-1718638750000z^{7}w^{2}t^{2}-4722064250000z^{7}wt^{3}-2764891125000z^{7}t^{4}+10591858860000z^{5}w^{2}t^{4}+10843357680000z^{5}wt^{5}+1846617170000z^{5}t^{6}-6100480798050z^{3}w^{2}t^{6}-31255192219950z^{3}wt^{7}-23988025037900z^{3}t^{8}-10240000000zw^{10}-153600000000zw^{9}t-808960000000zw^{8}t^{2}-2140160000000zw^{7}t^{3}-3665918950000zw^{6}t^{4}-4270440665000zw^{5}t^{5}-4042319375400zw^{4}t^{6}-1237533654990zw^{3}t^{7}-19903742556866zw^{2}t^{8}-43442430733293zwt^{9}-18686596637537zt^{10}}{t(50000000xz^{8}t+251000000xz^{6}t^{3}-954260000xz^{4}t^{5}-1005773675xz^{2}t^{7}-278073596xw^{2}t^{7}-625899192xwt^{8}-74276798xt^{9}+50000000yz^{8}t+146750000yz^{6}t^{3}-936040000yz^{4}t^{5}-784456075yz^{2}t^{7}+431414394yt^{9}+3125000z^{9}w+3125000z^{9}t+4250000z^{7}w^{2}t+88750000z^{7}wt^{2}+83875000z^{7}t^{3}-99060000z^{5}w^{2}t^{3}-38860000z^{5}wt^{4}+63000000z^{5}t^{5}-612916250z^{3}w^{2}t^{5}-1244417900z^{3}wt^{6}-684647650z^{3}t^{7}-39000zw^{5}t^{4}+4181400zw^{4}t^{5}+70647050zw^{3}t^{6}-425189766zw^{2}t^{7}-877296653zwt^{8}-371268277zt^{9})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.72.3.e.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{5}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y+X^{4}Z-X^{2}Y^{2}Z-6X^{2}YZ^{2}-6X^{2}Z^{3}+5YZ^{4}+5Z^{5} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.72.3.e.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{1}{25}y^{2}wt-\frac{3}{25}y^{2}t^{2}+\frac{2}{625}t^{4}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}t$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.72.1-10.a.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.288.5-20.b.2.23 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-20.c.1.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-20.f.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.f.2.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-20.g.2.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.i.1.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.r.1.3 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.u.2.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.7-20.h.2.5 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.h.2.7 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.j.2.5 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.j.2.16 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.k.1.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-20.k.1.8 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-20.m.2.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-20.m.2.7 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.x.1.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.x.1.14 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bb.2.6 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bb.2.12 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.be.2.2 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{2}\cdot2$ |
| 40.288.7-40.be.2.16 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{2}\cdot2$ |
| 40.288.7-40.bi.1.6 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bi.1.10 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.720.19-20.q.1.6 | $40$ | $5$ | $5$ | $19$ | $0$ | $1^{6}\cdot2^{5}$ |
| 120.288.5-60.bh.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bi.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bt.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bu.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dx.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ea.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fh.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fk.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-60.em.2.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.em.2.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.eo.1.2 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.eo.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.et.2.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.et.2.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.ev.2.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-60.ev.2.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcw.1.11 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcw.1.31 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdc.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdc.1.28 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdp.1.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdp.1.31 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdv.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdv.1.28 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.432.15-60.w.2.17 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-140.q.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.s.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.u.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-140.w.2.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.by.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ce.1.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ck.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cq.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-140.s.2.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.s.2.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.t.2.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.t.2.9 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.w.2.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.w.2.13 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.x.2.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-140.x.2.9 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.ck.1.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.ck.1.26 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cn.2.10 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cn.2.18 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cw.2.2 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cw.2.26 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cz.1.10 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cz.1.18 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |