Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-5232187x+2915000634\)
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(homogenize, simplify) |
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\(y^2z=x^3-5232187xz^2+2915000634z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5232187x+2915000634\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1973, 16500\right) \) | $1.3995830641119349562455230504$ | $\infty$ |
| \( \left(598, 0\right) \) | $0$ | $2$ |
| \( \left(1929, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1973:16500:1]\) | $1.3995830641119349562455230504$ | $\infty$ |
| \([598:0:1]\) | $0$ | $2$ |
| \([1929:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1973, 16500\right) \) | $1.3995830641119349562455230504$ | $\infty$ |
| \( \left(598, 0\right) \) | $0$ | $2$ |
| \( \left(1929, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-2527, 0\right) \), \((-2427,\pm 36300)\), \( \left(598, 0\right) \), \( \left(1929, 0\right) \), \((1973,\pm 16500)\), \((136723,\pm 50547750)\)
\([-2527:0:1]\), \([-2427:\pm 36300:1]\), \([598:0:1]\), \([1929:0:1]\), \([1973:\pm 16500:1]\), \([136723:\pm 50547750:1]\)
\( \left(-2527, 0\right) \), \((-2427,\pm 36300)\), \( \left(598, 0\right) \), \( \left(1929, 0\right) \), \((1973,\pm 16500)\), \((136723,\pm 50547750)\)
Invariants
| Conductor: | $N$ | = | \( 490160 \) | = | $2^{4} \cdot 5 \cdot 11 \cdot 557$ |
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| Minimal Discriminant: | $\Delta$ | = | $5496250286890000000000$ | = | $2^{10} \cdot 5^{10} \cdot 11^{6} \cdot 557^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{15469402078671548025924}{5367431920791015625} \) | = | $2^{2} \cdot 3^{3} \cdot 5^{-10} \cdot 11^{-6} \cdot 557^{-2} \cdot 5232187^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8731643053834807943447299149$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2955416549168597031637031470$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0394653197594943$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.428519724474745$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3995830641119349562455230504$ |
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| Real period: | $\Omega$ | ≈ | $0.12447545562040181820860178647$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 480 $ = $ 2^{2}\cdot( 2 \cdot 5 )\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.2264121875179345695423130977 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.226412188 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.124475 \cdot 1.399583 \cdot 480}{4^2} \\ & \approx 5.226412188\end{aligned}$$
Modular invariants
Modular form 490160.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18439680 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $5$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $557$ | $2$ | $I_{2}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 245080 = 2^{3} \cdot 5 \cdot 11 \cdot 557 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 245077 & 4 \\ 245076 & 5 \end{array}\right),\left(\begin{array}{rr} 122541 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 200521 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 61267 & 245076 \\ 245072 & 245069 \end{array}\right),\left(\begin{array}{rr} 147051 & 2 \\ 147046 & 245079 \end{array}\right),\left(\begin{array}{rr} 148721 & 4 \\ 52362 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[245080])$ is a degree-$19480680525791232000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/245080\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | good | $2$ | \( 44560 = 2^{4} \cdot 5 \cdot 557 \) |
| $5$ | split multiplicative | $6$ | \( 98032 = 2^{4} \cdot 11 \cdot 557 \) |
| $11$ | split multiplicative | $12$ | \( 44560 = 2^{4} \cdot 5 \cdot 557 \) |
| $557$ | nonsplit multiplicative | $558$ | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 490160.h
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 245080.f3, its twist by $-4$.
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.