Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $50$ | ||
| Index: | $60$ | $\PSL_2$-index: | $30$ | ||||
| Genus: | $2 = 1 + \frac{ 30 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $5^{2}\cdot20$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 20A2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.60.2.8 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&14\\32&23\end{bmatrix}$, $\begin{bmatrix}7&35\\16&13\end{bmatrix}$, $\begin{bmatrix}11&8\\4&31\end{bmatrix}$, $\begin{bmatrix}35&19\\28&15\end{bmatrix}$, $\begin{bmatrix}39&15\\8&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.30.2.c.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $12288$ |
Jacobian
| Conductor: | $2^{2}\cdot5^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}$ |
| Newforms: | 50.2.a.b$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{3} - 2 x^{2} w + 2 x y^{2} + 2 x z^{2} - y z w $ |
| $=$ | $3 x^{2} w - x w^{2} + y^{2} w + 2 y z w$ | |
| $=$ | $x^{3} + x^{2} w - x y^{2} + 2 x y z - 2 x z^{2} + y z w$ | |
| $=$ | $3 x^{2} z - x z w + y^{2} z + 2 y z^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{5} - 6 x^{4} y + x^{3} y^{2} + 10 x^{3} z^{2} - 3 x^{2} y z^{2} - x y^{2} z^{2} + 5 x z^{4} + y z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{2} + x\right) y $ | $=$ | $ x^{6} - 3x^{5} + 7x^{4} - 10x^{3} + 7x^{2} - 3x + 1 $ |
Rational points
This modular curve has 3 rational cusps and 2 rational CM points, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Plane model | Weierstrass model | Embedded model | |
|---|---|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | |||||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(0:0:1)$ | $(1:-1:0)$ | $(0:-2:1:0)$ |
| 32.a1 | $-16$ | $287496$ | $= 2^{3} \cdot 3^{3} \cdot 11^{3}$ | $12.569$ | $(1:3:1)$, $(-1:-3:1)$ | $(0:-1:1)$, $(1:0:1)$ | $(1/3:1/3:-1/6:1)$, $(1/3:-1/3:1/6:1)$ |
Maps to other modular curves
$j$-invariant map of degree 30 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(3z^{2}-w^{2})^{3}}{z^{4}(2z-w)(2z+w)}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.30.2.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{5}-6X^{4}Y+X^{3}Y^{2}+10X^{3}Z^{2}-3X^{2}YZ^{2}-XY^{2}Z^{2}+5XZ^{4}+YZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.30.2.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle \frac{1}{2}xy-\frac{1}{2}y^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{3}{4}x^{4}y^{2}-\frac{3}{8}x^{3}y^{3}+\frac{1}{4}x^{3}y^{2}w+\frac{1}{8}x^{2}y^{4}-\frac{1}{8}xy^{5}-\frac{1}{4}xy^{4}w+\frac{1}{8}y^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle xy$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
| 8.12.0-4.c.1.3 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0-4.c.1.3 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.120.4-20.b.1.11 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.f.1.2 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-20.h.1.3 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 40.120.4-20.k.1.1 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-20.l.1.1 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 40.120.4-40.w.1.3 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 40.120.4-40.be.1.1 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 40.120.4-40.bh.1.4 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bk.1.8 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bk.1.9 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bl.1.5 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bl.1.24 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bm.1.7 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 40.120.4-40.bm.1.10 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 40.120.4-40.bn.1.2 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bn.1.15 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bo.1.2 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 40.120.4-40.bo.1.15 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 40.120.4-40.bp.1.7 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bp.1.10 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 40.120.4-40.bq.1.1 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 40.120.4-40.bq.1.16 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 40.120.4-40.br.1.8 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 40.120.4-40.br.1.9 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 40.180.4-20.c.1.14 | $40$ | $3$ | $3$ | $4$ | $0$ | $1^{2}$ |
| 40.240.5-20.m.1.6 | $40$ | $4$ | $4$ | $5$ | $0$ | $1^{3}$ |
| 120.120.4-60.k.1.2 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-60.l.1.4 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-60.o.1.4 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-60.p.1.4 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.be.1.7 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bh.1.3 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bq.1.2 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bt.1.4 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bw.1.16 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bw.1.17 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bx.1.14 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bx.1.19 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.by.1.1 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.by.1.32 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bz.1.3 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.bz.1.30 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.ca.1.3 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.ca.1.30 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.cb.1.1 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.cb.1.32 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.cc.1.14 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.cc.1.19 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.cd.1.16 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.120.4-120.cd.1.17 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.180.7-60.c.1.11 | $120$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 120.240.8-60.o.1.23 | $120$ | $4$ | $4$ | $8$ | $?$ | not computed |
| 280.120.4-140.k.1.2 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-140.l.1.2 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-140.o.1.3 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-140.p.1.3 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.be.1.4 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bh.1.5 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bq.1.6 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bt.1.3 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bw.1.11 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bw.1.22 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bx.1.14 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bx.1.19 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.by.1.3 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.by.1.30 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bz.1.6 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.bz.1.27 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.ca.1.6 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.ca.1.27 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.cb.1.3 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.cb.1.30 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.cc.1.14 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.cc.1.19 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.cd.1.11 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.120.4-280.cd.1.22 | $280$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 280.480.18-140.c.1.31 | $280$ | $8$ | $8$ | $18$ | $?$ | not computed |