Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-34790720x-78984748304\)
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(homogenize, simplify) |
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\(y^2z=x^3-34790720xz^2-78984748304z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-34790720x-78984748304\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 425104 \) | = | $2^{4} \cdot 163^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-17738739712$ | = | $-1 \cdot 2^{12} \cdot 163^{3} $ |
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| j-invariant: | $j$ | = | \( -262537412640768000 \) | = | $-1 \cdot 2^{18} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3} \cdot 29^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-163})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5442745370249491192264004706$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.57768980626331322629293525761$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2036213229011967$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.915721346623763$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.031081624175576129698174098151$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.24865299340460903758539278521 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 0.248652993 \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{4 \cdot 0.031082 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 0.248652993\end{aligned}$$
Modular invariants
Modular form 425104.2.a.g
\( q - 3 q^{9} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 4274496 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $163$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 | $2$ | \( 163 \) |
| $163$ | additive | $6806$ | \( 16 = 2^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
163.
Its isogeny class 425104.g
consists of 2 curves linked by isogenies of
degree 163.
Twists
The minimal quadratic twist of this elliptic curve is 26569.a2, its twist by $-4$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.