Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+724768x-357175820\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+724768xz^2-357175820z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+58706181x-260557291350\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 479408 \) | = | $2^{4} \cdot 19^{2} \cdot 83$ |
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| Discriminant: | $\Delta$ | = | $-79552352751814795264$ | = | $-1 \cdot 2^{13} \cdot 19^{8} \cdot 83^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{605245247}{1143574} \) | = | $2^{-1} \cdot 19 \cdot 83^{-3} \cdot 317^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5032155978202996822099953062$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.15289090218393926721325510318$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8984436837305189$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.04516301580859$ | |||
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: | $\Omega$ | ≈ | $0.10083141350210105048316547125$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2^{2}\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.8399078481008504231919426198 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 4.839907848 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.100831 \cdot 1.000000 \cdot 12}{1^2} \\ & \approx 4.839907848\end{aligned}$$
Modular invariants
Modular form 479408.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15266880 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $19$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $83$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 \( 664 = 2^{3} \cdot 83 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 417 & 2 \\ 417 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 663 & 0 \end{array}\right),\left(\begin{array}{rr} 663 & 2 \\ 662 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 333 & 2 \\ 333 & 3 \end{array}\right),\left(\begin{array}{rr} 167 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[664])$ is a degree-$36003631104$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/664\Z)$.
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 | $4$ | \( 29963 = 19^{2} \cdot 83 \) |
| $3$ | good | $2$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $146$ | \( 1328 = 2^{4} \cdot 83 \) |
| $83$ | nonsplit multiplicative | $84$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 479408.h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 59926.c1, its twist by $76$.
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 $0$.