Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-6457216x+6315489838\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-6457216xz^2+6315489838z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8368551315x+294680599547310\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(23213/16, 33473/64)$ | $8.5464072635039337821330746028$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462550 \) | = | $2 \cdot 5^{2} \cdot 11 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-2222487863552607200$ | = | $-1 \cdot 2^{5} \cdot 5^{2} \cdot 11^{5} \cdot 29^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{2002311132699145}{149455328} \) | = | $-1 \cdot 2^{-5} \cdot 5 \cdot 11^{-5} \cdot 29^{-1} \cdot 73709^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5700788618892360990366039268$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.61819129482364902301150802175$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9274124342337671$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.496593473375928$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.5464072635039337821330746028$ |
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| Real period: | $\Omega$ | ≈ | $0.24736119281801518870079217540$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.2280989900177642148129938118 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.228098990 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.247361 \cdot 8.546407 \cdot 2}{1^2} \\ & \approx 4.228098990\end{aligned}$$
Modular invariants
Modular form 462550.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11088000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $11$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $29$ | $2$ | $I_{1}^{*}$ | 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 |
|---|---|---|
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12760 = 2^{3} \cdot 5 \cdot 11 \cdot 29 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 3191 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10441 & 10 \\ 1165 & 51 \end{array}\right),\left(\begin{array}{rr} 6381 & 10 \\ 6385 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 3191 & 6390 \\ 0 & 10847 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 12705 & 12641 \end{array}\right),\left(\begin{array}{rr} 10119 & 12750 \\ 12315 & 12709 \end{array}\right),\left(\begin{array}{rr} 12751 & 10 \\ 12750 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[12760])$ is a degree-$138293084160000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12760\Z)$.
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 | $4$ | \( 231275 = 5^{2} \cdot 11 \cdot 29^{2} \) |
| $5$ | additive | $10$ | \( 841 = 29^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 42050 = 2 \cdot 5^{2} \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 462550v
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 15950n1, its twist by $29$.
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.