Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-16679x-782235\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-16679xz^2-782235z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-21616659x-36171709650\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1458, 54741\right) \) | $2.8800637431231633757874937791$ | $\infty$ |
| \( \left(-90, 45\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1458:54741:1]\) | $2.8800637431231633757874937791$ | $\infty$ |
| \([-90:45:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(52503, 11981520\right) \) | $2.8800637431231633757874937791$ | $\infty$ |
| \( \left(-3225, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-90, 45\right) \), \( \left(1458, 54741\right) \), \( \left(1458, -56199\right) \)
\([-90:45:1]\), \([1458:54741:1]\), \([1458:-56199:1]\)
\( \left(-3225, 0\right) \), \((52503,\pm 11981520)\)
Invariants
| Conductor: | $N$ | = | \( 454854 \) | = | $2 \cdot 3 \cdot 41 \cdot 43^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $37321327441296$ | = | $2^{4} \cdot 3^{2} \cdot 41 \cdot 43^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{81182737}{5904} \) | = | $2^{-4} \cdot 3^{-2} \cdot 41^{-1} \cdot 433^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3505950284596773582546312137$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.53000502938710385348179004297$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9582556017330255$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.1302006064068193$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.8800637431231633757874937791$ |
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| Real period: | $\Omega$ | ≈ | $0.42202921990286280248438256850$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.8618842191231507309713587979 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.861884219 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.422029 \cdot 2.880064 \cdot 16}{2^2} \\ & \approx 4.861884219\end{aligned}$$
Modular invariants
Modular form 454854.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1935360 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $41$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $43$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 |
|---|---|---|---|
| $2$ | 2B | 8.12.0.11 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14104 = 2^{3} \cdot 41 \cdot 43 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 14098 & 14099 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 11611 & 7998 \\ 13330 & 2667 \end{array}\right),\left(\begin{array}{rr} 14097 & 8 \\ 14096 & 9 \end{array}\right),\left(\begin{array}{rr} 3140 & 11481 \\ 2967 & 990 \end{array}\right),\left(\begin{array}{rr} 4473 & 4472 \\ 13502 & 2839 \end{array}\right),\left(\begin{array}{rr} 8855 & 0 \\ 0 & 14103 \end{array}\right)$.
The torsion field $K:=\Q(E[14104])$ is a degree-$294254302003200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14104\Z)$.
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 | $4$ | \( 75809 = 41 \cdot 43^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 151618 = 2 \cdot 41 \cdot 43^{2} \) |
| $41$ | split multiplicative | $42$ | \( 11094 = 2 \cdot 3 \cdot 43^{2} \) |
| $43$ | additive | $926$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 454854d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 246e1, its twist by $-43$.
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.