Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-4719x-23958\)
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(homogenize, simplify) |
\(y^2z=x^3-4719xz^2-23958z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-4719x-23958\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(-66, 0)$ | $0$ | $2$ |
Integral points
\( \left(-66, 0\right) \)
Invariants
Conductor: | $N$ | = | \( 400752 \) | = | $2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 23$ |
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Discriminant: | $\Delta$ | = | $6477620675328$ | = | $2^{8} \cdot 3^{3} \cdot 11^{6} \cdot 23^{2} $ |
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j-invariant: | $j$ | = | \( \frac{949104}{529} \) | = | $2^{4} \cdot 3^{3} \cdot 13^{3} \cdot 23^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1486001168689094751235719312$ |
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.78709871207060009270103258132$ |
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$abc$ quality: | $Q$ | ≈ | $1.027047455683347$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.867326442501543$ |
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: | $\Omega$ | ≈ | $0.61812762251704792671116915289$ |
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Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot2 $ |
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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Special value: | $ L(E,1)$ | ≈ | $2.4725104900681917068446766116 $ |
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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.472510490 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.618128 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.472510490\end{aligned}$$
Modular invariants
Modular form 400752.2.a.fh
For more coefficients, see the Downloads section to the right.
Modular degree: | 491520 |
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$ \Gamma_0(N) $-optimal: | no | |
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_0^{*}$ | additive | -1 | 4 | 8 | 0 |
$3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
$11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
$23$ | $2$ | $I_{2}$ | 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 |
---|---|---|
$2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 276 = 2^{2} \cdot 3 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 208 & 73 \\ 69 & 208 \end{array}\right),\left(\begin{array}{rr} 97 & 4 \\ 194 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 273 & 4 \\ 272 & 5 \end{array}\right),\left(\begin{array}{rr} 188 & 1 \\ 91 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[276])$ is a degree-$102592512$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/276\Z)$.
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 |
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$2$ | additive | $2$ | \( 363 = 3 \cdot 11^{2} \) |
$3$ | additive | $6$ | \( 44528 = 2^{4} \cdot 11^{2} \cdot 23 \) |
$11$ | additive | $62$ | \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \) |
$23$ | split multiplicative | $24$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 400752fh
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 828b2, its twist by $-132$.
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 $0$.