Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-3920408x+2739831688\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-3920408xz^2+2739831688z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-317553075x+1998289959750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1402, 1078)$ | $3.4103669806233326235016859734$ | $\infty$ |
| $(863, 0)$ | $0$ | $2$ |
Integral points
\((-2262,\pm 6250)\), \( \left(863, 0\right) \), \((1402,\pm 1078)\), \((839919,\pm 769759684)\)
Invariants
| Conductor: | $N$ | = | \( 431200 \) | = | $2^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11$ |
|
| Discriminant: | $\Delta$ | = | $610348305875000000000$ | = | $2^{9} \cdot 5^{12} \cdot 7^{9} \cdot 11^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{7080100070408}{648484375} \) | = | $2^{3} \cdot 5^{-6} \cdot 7^{-3} \cdot 11^{-2} \cdot 9601^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7274490545289942202050761169$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.42991463836432839828909598747$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8994849606031108$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.405524913651446$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4103669806233326235016859734$ |
|
| Real period: | $\Omega$ | ≈ | $0.15846106947879472934079025200$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.3232831925179304744237463996 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.323283193 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.158461 \cdot 3.410367 \cdot 32}{2^2} \\ & \approx 4.323283193\end{aligned}$$
Modular invariants
Modular form 431200.2.a.ba
For more coefficients, see the Downloads section to the right.
| Modular degree: | 21233664 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 5 | 9 | 0 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $7$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 3077 & 4 \\ 3076 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 617 & 4 \\ 1234 & 9 \end{array}\right),\left(\begin{array}{rr} 1322 & 1 \\ 879 & 0 \end{array}\right),\left(\begin{array}{rr} 2521 & 4 \\ 1962 & 9 \end{array}\right),\left(\begin{array}{rr} 1156 & 1929 \\ 385 & 2696 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1539 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[3080])$ is a degree-$1634992128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3080\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 17248 = 2^{5} \cdot 7^{2} \cdot 11 \) |
| $7$ | additive | $32$ | \( 8800 = 2^{5} \cdot 5^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 39200 = 2^{5} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 431200ba
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 12320n2, its twist by $-35$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.