Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+1217087x+6372602137\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+1217087xz^2+6372602137z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1577344077x+297296465139342\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(8882, 842897\right) \) | $8.8810273706305994286708979852$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([8882:842897:1]\) | $8.8810273706305994286708979852$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(319767, 183025008\right) \) | $8.8810273706305994286708979852$ | $\infty$ |
Integral points
\( \left(8882, 842897\right) \), \( \left(8882, -851779\right) \)
\([8882:842897:1]\), \([8882:-851779:1]\)
\((319767,\pm 183025008)\)
Invariants
| Conductor: | $N$ | = | \( 490110 \) | = | $2 \cdot 3 \cdot 5 \cdot 17 \cdot 31^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-17656136656123885361460$ | = | $-1 \cdot 2^{2} \cdot 3^{6} \cdot 5 \cdot 17^{5} \cdot 31^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{233773067111}{20701515060} \) | = | $2^{-2} \cdot 3^{-6} \cdot 5^{-1} \cdot 17^{-5} \cdot 31 \cdot 37^{3} \cdot 53^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9482525332289097076084504768$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.65892773023881221032234092710$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9500087706541115$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.478080381220144$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.8810273706305994286708979852$ |
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| Real period: | $\Omega$ | ≈ | $0.094110280734216439107164401963$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3431839162332231219108804848 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.343183916 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.094110 \cdot 8.881027 \cdot 4}{1^2} \\ & \approx 3.343183916\end{aligned}$$
Modular invariants
Modular form 490110.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 48211200 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $17$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $31$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
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 \( 340 = 2^{2} \cdot 5 \cdot 17 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 339 & 0 \end{array}\right),\left(\begin{array}{rr} 339 & 2 \\ 338 & 3 \end{array}\right),\left(\begin{array}{rr} 171 & 2 \\ 171 & 3 \end{array}\right),\left(\begin{array}{rr} 241 & 2 \\ 241 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 137 & 2 \\ 137 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[340])$ is a degree-$1804861440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/340\Z)$.
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 | $4$ | \( 81685 = 5 \cdot 17 \cdot 31^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 163370 = 2 \cdot 5 \cdot 17 \cdot 31^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 5766 = 2 \cdot 3 \cdot 31^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 28830 = 2 \cdot 3 \cdot 5 \cdot 31^{2} \) |
| $31$ | additive | $362$ | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 490110c consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 490110x1, its twist by $-31$.
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.