Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-663567x-205884659\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-663567xz^2-205884659z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-10617075x-13187235250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-495, 1301)$ | $3.8498325879066568213453094644$ | $\infty$ |
| $(-2029/4, 2029/8)$ | $0$ | $2$ |
Integral points
\( \left(-495, 1301\right) \), \( \left(-495, -806\right) \), \( \left(8399, 761738\right) \), \( \left(8399, -770137\right) \)
Invariants
| Conductor: | $N$ | = | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $358351500937500000$ | = | $2^{5} \cdot 3^{3} \cdot 5^{10} \cdot 7^{6} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{651038076963}{7220000} \) | = | $2^{-5} \cdot 3^{12} \cdot 5^{-4} \cdot 19^{-2} \cdot 107^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1826804547674805145434197565$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.13035335185574625184155240894$ |
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| $abc$ quality: | $Q$ | ≈ | $1.041120842060056$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.003687129650014$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.8498325879066568213453094644$ |
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| Real period: | $\Omega$ | ≈ | $0.16738608755549564439875999585$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.1552673170667523265872287736 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.155267317 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.167386 \cdot 3.849833 \cdot 32}{2^2} \\ & \approx 5.155267317\end{aligned}$$
Modular invariants
Modular form 418950.2.a.f
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8847360 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $2$ | $I_{2}$ | 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 \( 456 = 2^{3} \cdot 3 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 4 \\ 194 & 9 \end{array}\right),\left(\begin{array}{rr} 172 & 289 \\ 57 & 400 \end{array}\right),\left(\begin{array}{rr} 453 & 4 \\ 452 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 227 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 308 & 1 \\ 151 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[456])$ is a degree-$756449280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/456\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 46550 = 2 \cdot 5^{2} \cdot 7^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $26$ | \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 22050 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 418950.f
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1710.a1, its twist by $105$.
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.