Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+2539572x-6860596624\)
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(homogenize, simplify) |
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\(y^2z=x^3+2539572xz^2-6860596624z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2539572x-6860596624\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(10990, 1161216)$ | $2.6814596111967529389113392547$ | $\infty$ |
Integral points
\((10990,\pm 1161216)\)
Invariants
| Conductor: | $N$ | = | \( 479808 \) | = | $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-21381525586736320610304$ | = | $-1 \cdot 2^{31} \cdot 3^{15} \cdot 7^{4} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{3947714094191}{46599266304} \) | = | $2^{-13} \cdot 3^{-9} \cdot 7^{5} \cdot 17^{-2} \cdot 617^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9656476349917529714530829978$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.72798400346600905992782794934$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1048697407102008$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.496646329888461$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.6814596111967529389113392547$ |
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| Real period: | $\Omega$ | ≈ | $0.059470174674817179226874285009$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot2\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.6544098301442363144313238551 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.654409830 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.059470 \cdot 2.681460 \cdot 48}{1^2} \\ & \approx 7.654409830\end{aligned}$$
Modular invariants
Modular form 479808.2.a.fy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 25878528 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{21}^{*}$ | additive | 1 | 6 | 31 | 13 |
| $3$ | $2$ | $I_{9}^{*}$ | additive | -1 | 2 | 15 | 9 |
| $7$ | $3$ | $IV$ | additive | 1 | 2 | 4 | 0 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 label 24.2.0.b.1, level \( 24 = 2^{3} \cdot 3 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17 & 2 \\ 17 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 2 \\ 22 & 3 \end{array}\right),\left(\begin{array}{rr} 13 & 2 \\ 13 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 23 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$36864$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 | $4$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 53312 = 2^{6} \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $20$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 479808fy consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 4998bi1, its twist by $-24$.
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.