Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-9369230x+10819331772\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-9369230xz^2+10819331772z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-149907675x+692287325750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-285566/81, 4710745/729)$ | $11.357709281607022144590981227$ | $\infty$ |
| $(1559, -780)$ | $0$ | $2$ |
Integral points
\( \left(1559, -780\right) \)
Invariants
| Conductor: | $N$ | = | \( 416025 \) | = | $3^{2} \cdot 5^{2} \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $2089924110319953515625$ | = | $3^{9} \cdot 5^{8} \cdot 43^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{1263214441}{29025} \) | = | $3^{-3} \cdot 5^{-2} \cdot 23^{3} \cdot 43^{-1} \cdot 47^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8772770089533311261427200581$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.35734814944455511859170348365$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8516913279930017$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.619733696075128$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $11.357709281607022144590981227$ |
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| Real period: | $\Omega$ | ≈ | $0.14665866218241431386234301136$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6628257947891032927295741943 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.662825795 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.146659 \cdot 11.357709 \cdot 16}{2^2} \\ & \approx 6.662825795\end{aligned}$$
Modular invariants
Modular form 416025.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 25546752 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $43$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 \( 2580 = 2^{2} \cdot 3 \cdot 5 \cdot 43 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1382 & 1 \\ 599 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 862 & 1 \\ 859 & 0 \end{array}\right),\left(\begin{array}{rr} 649 & 1936 \\ 644 & 1935 \end{array}\right),\left(\begin{array}{rr} 2577 & 4 \\ 2576 & 5 \end{array}\right),\left(\begin{array}{rr} 517 & 4 \\ 1034 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[2580])$ is a degree-$615165788160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2580\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $6$ | \( 46225 = 5^{2} \cdot 43^{2} \) |
| $5$ | additive | $18$ | \( 16641 = 3^{2} \cdot 43^{2} \) |
| $43$ | additive | $968$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 416025.t
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 645.e1, its twist by $645$.
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.