Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+9514877x-5564089390\)
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(homogenize, simplify) |
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\(y^2z=x^3+9514877xz^2-5564089390z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9514877x-5564089390\)
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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: | $\Delta$ | = | $-68504519763406515571712$ | = | $-1 \cdot 2^{10} \cdot 19^{8} \cdot 83^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{1977478112299644}{1421993672123} \) | = | $2^{2} \cdot 3^{3} \cdot 19^{-2} \cdot 83^{-5} \cdot 26357^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0701537144840851715676943384$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0203115744342438503821538546$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9537332812228135$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.5731880076645774$ | |||
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: | $\Omega$ | ≈ | $0.061751870520050687145594433155$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $17.784538709774597897931196749 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $36$ = $6^2$ (exact) |
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BSD formula
$$\begin{aligned} 17.784538710 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{36 \cdot 0.061752 \cdot 1.000000 \cdot 8}{1^2} \\ & \approx 17.784538710\end{aligned}$$
Modular invariants
Modular form 479408.2.a.bd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 108979200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $19$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $83$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
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 \( 166 = 2 \cdot 83 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 165 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 165 & 2 \\ 164 & 3 \end{array}\right),\left(\begin{array}{rr} 85 & 2 \\ 85 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[166])$ is a degree-$140639184$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/166\Z)$.
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 | $2$ | \( 29963 = 19^{2} \cdot 83 \) |
| $5$ | good | $2$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1328 = 2^{4} \cdot 83 \) |
| $83$ | nonsplit multiplicative | $84$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 479408.bd consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 12616.e1, 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 $0$.