Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+654x-30815\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+654xz^2-30815z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+10469x-1961674\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{29611}{324}, \frac{4901329}{5832}\right) \) | $10.502014379242758114505225969$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([532998:4901329:5832]\) | $10.502014379242758114505225969$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{29530}{81}, \frac{5170744}{729}\right) \) | $10.502014379242758114505225969$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 406486 \) | = | $2 \cdot 19^{2} \cdot 563$ |
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| Minimal Discriminant: | $\Delta$ | = | $-423789296048$ | = | $-1 \cdot 2^{4} \cdot 19^{6} \cdot 563 $ |
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| j-invariant: | $j$ | = | \( \frac{658503}{9008} \) | = | $2^{-4} \cdot 3^{3} \cdot 29^{3} \cdot 563^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.91178412178264191418691245167$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.56043536780057831581760126427$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8061140498221601$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.646782053086727$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.502014379242758114505225969$ |
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| Real period: | $\Omega$ | ≈ | $0.46212898628684377905342744346$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $19.413141036197250990059998577 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 19.413141036 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.462129 \cdot 10.502014 \cdot 4}{1^2} \\ & \approx 19.413141036\end{aligned}$$
Modular invariants
Modular form 406486.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1188000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $19$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $563$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 \( 1126 = 2 \cdot 563 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1125 & 2 \\ 1124 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1125 & 0 \end{array}\right),\left(\begin{array}{rr} 565 & 2 \\ 565 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[1126])$ is a degree-$300871731024$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1126\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 203243 = 19^{2} \cdot 563 \) |
| $19$ | additive | $182$ | \( 1126 = 2 \cdot 563 \) |
| $563$ | nonsplit multiplicative | $564$ | \( 722 = 2 \cdot 19^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 406486.d consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1126.a1, its twist by $-19$.
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.