Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+4335x-87206\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+4335xz^2-87206z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+69360x-5581168\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(68, 722)$ | $1.8879485469665324532897503725$ | $\infty$ |
Integral points
\( \left(68, 722\right) \), \( \left(68, -723\right) \)
Invariants
| Conductor: | $N$ | = | \( 418761 \) | = | $3^{2} \cdot 7 \cdot 17^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-8499007007883$ | = | $-1 \cdot 3^{7} \cdot 7 \cdot 17^{6} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{512000}{483} \) | = | $2^{12} \cdot 3^{-1} \cdot 5^{3} \cdot 7^{-1} \cdot 23^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1680836033918450129387575254$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.79782921297031787288363240200$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7434793360740976$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.837920110130763$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8879485469665324532897503725$ |
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| Real period: | $\Omega$ | ≈ | $0.40161901626723743881352626896$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.0329441527834363325014001978 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.032944153 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.401619 \cdot 1.887949 \cdot 4}{1^2} \\ & \approx 3.032944153\end{aligned}$$
Modular invariants
Modular form 418761.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 788480 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $23$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 965 & 0 \end{array}\right),\left(\begin{array}{rr} 323 & 2 \\ 323 & 3 \end{array}\right),\left(\begin{array}{rr} 925 & 2 \\ 925 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 965 & 2 \\ 964 & 3 \end{array}\right),\left(\begin{array}{rr} 829 & 2 \\ 829 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[966])$ is a degree-$77559939072$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/966\Z)$.
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 | $8$ | \( 46529 = 7 \cdot 17^{2} \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 59823 = 3^{2} \cdot 17^{2} \cdot 23 \) |
| $17$ | additive | $146$ | \( 1449 = 3^{2} \cdot 7 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 18207 = 3^{2} \cdot 7 \cdot 17^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 418761.a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 483.a1, its twist by $-51$.
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.