Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-4381182x+3527694045\)
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(homogenize, simplify) |
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\(y^2z=x^3-4381182xz^2+3527694045z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4381182x+3527694045\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(53502/49, 2319057/343)$ | $8.2744537334844983729921582040$ | $\infty$ |
| $(1185, 0)$ | $0$ | $2$ |
Integral points
\( \left(1185, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 449352 \) | = | $2^{3} \cdot 3^{2} \cdot 79^{2}$ |
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| Discriminant: | $\Delta$ | = | $6047848649193081552$ | = | $2^{4} \cdot 3^{9} \cdot 79^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{121485312}{79} \) | = | $2^{11} \cdot 3^{3} \cdot 13^{3} \cdot 79^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5438139205027041213646973116$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.69591828241853733074062009399$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8450331076841464$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.417180843765908$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.2744537334844983729921582040$ |
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| Real period: | $\Omega$ | ≈ | $0.23659884005171299583366719191$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9154523108079965289547755314 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.915452311 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.236599 \cdot 8.274454 \cdot 8}{2^2} \\ & \approx 3.915452311\end{aligned}$$
Modular invariants
Modular form 449352.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 22164480 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $79$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 948 = 2^{2} \cdot 3 \cdot 79 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 713 & 238 \\ 236 & 711 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 945 & 4 \\ 944 & 5 \end{array}\right),\left(\begin{array}{rr} 638 & 1 \\ 155 & 0 \end{array}\right),\left(\begin{array}{rr} 320 & 1 \\ 631 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[948])$ is a degree-$14765137920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/948\Z)$.
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 | $2$ | \( 18723 = 3 \cdot 79^{2} \) |
| $3$ | additive | $2$ | \( 49928 = 2^{3} \cdot 79^{2} \) |
| $79$ | additive | $3200$ | \( 72 = 2^{3} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 449352a
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 5688d1, its twist by $-79$.
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.