Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1342170x+1770936372\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1342170xz^2+1770936372z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-21474723x+113318453086\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3108, 164622\right) \) | $5.0570831234546929406227054398$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([3108:164622:1]\) | $5.0570831234546929406227054398$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(12431, 1329408\right) \) | $5.0570831234546929406227054398$ | $\infty$ |
Integral points
\( \left(3108, 164622\right) \), \( \left(3108, -167730\right) \)
\([3108:164622:1]\), \([3108:-167730:1]\)
\((12431,\pm 1329408)\)
Invariants
| Conductor: | $N$ | = | \( 410958 \) | = | $2 \cdot 3^{2} \cdot 17^{2} \cdot 79$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1199591740645359611904$ | = | $-1 \cdot 2^{12} \cdot 3^{12} \cdot 17^{8} \cdot 79 $ |
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| j-invariant: | $j$ | = | \( -\frac{52580846881}{235892736} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-6} \cdot 17 \cdot 31^{3} \cdot 47^{3} \cdot 79^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7295316397620880740470992959$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.29141659939055584151645359885$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9040232282440088$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.340959764896545$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.0570831234546929406227054398$ |
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| Real period: | $\Omega$ | ≈ | $0.13377995689766278927502539210$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.1184363474040007814321607455 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.118436347 \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 0.133780 \cdot 5.057083 \cdot 12}{1^2} \\ & \approx 8.118436347\end{aligned}$$
Modular invariants
Modular form 410958.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18330624 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $2$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $17$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $79$ | $1$ | $I_{1}$ | nonsplit 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 \( 158 = 2 \cdot 79 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 2 \\ 3 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 156 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 157 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[158])$ is a degree-$115352640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/158\Z)$.
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 | $4$ | \( 205479 = 3^{2} \cdot 17^{2} \cdot 79 \) |
| $3$ | additive | $6$ | \( 22831 = 17^{2} \cdot 79 \) |
| $17$ | additive | $114$ | \( 1422 = 2 \cdot 3^{2} \cdot 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 410958l consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 136986k1, 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.