Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-682651x+171406410\)
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(homogenize, simplify) |
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\(y^2z=x^3-682651xz^2+171406410z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-682651x+171406410\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(285, 0)$ | $0$ | $2$ |
| $(646, 0)$ | $0$ | $2$ |
Integral points
\( \left(-931, 0\right) \), \( \left(285, 0\right) \), \( \left(646, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 479408 \) | = | $2^{4} \cdot 19^{2} \cdot 83$ |
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| Discriminant: | $\Delta$ | = | $7667696650777329664$ | = | $2^{16} \cdot 19^{8} \cdot 83^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{182573756217}{39790864} \) | = | $2^{-4} \cdot 3^{3} \cdot 19^{-2} \cdot 31^{3} \cdot 61^{3} \cdot 83^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3373927885542218158505218032$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.17202611841105627642877596580$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9803417683840783$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.968929702428266$ | |||
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: | $\Omega$ | ≈ | $0.22120565839280862983583966332$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.44241131678561725967167932664 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.442411317 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.221206 \cdot 1.000000 \cdot 32}{4^2} \\ & \approx 0.442411317\end{aligned}$$
Modular invariants
Modular form 479408.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5529600 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | -1 | 4 | 16 | 4 |
| $19$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $83$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 4.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6308 = 2^{2} \cdot 19 \cdot 83 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3151 & 6304 \\ 6306 & 6305 \end{array}\right),\left(\begin{array}{rr} 663 & 6306 \\ 0 & 6307 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6305 & 4 \\ 6304 & 5 \end{array}\right),\left(\begin{array}{rr} 3575 & 2 \\ 1822 & 6307 \end{array}\right)$.
The torsion field $K:=\Q(E[6308])$ is a degree-$11543664222720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6308\Z)$.
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 | $2$ | \( 361 = 19^{2} \) |
| $19$ | additive | $200$ | \( 1328 = 2^{4} \cdot 83 \) |
| $83$ | split multiplicative | $84$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 479408.q
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3154.a2, 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 $0$.