Base field \(\Q(\sqrt{14}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$\left(-\frac{1}{2} : -\frac{1}{4} a - 1 : 1\right)$ | $1.0691085767972255605925653628188851268$ | $\infty$ |
$\left(6 : -9 a : 1\right)$ | $1.8919590029013207883469938309526819182$ | $\infty$ |
$\left(0 : -a : 1\right)$ | $0$ | $4$ |
Invariants
Conductor: | $\frak{N}$ | = | \((2a)\) | = | \((-a+4)^{3}\cdot(-2a+7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | $N(\frak{N})$ | = | \( 56 \) | = | \(2^{3}\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | $\Delta$ | = | $-28$ | ||
Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-28)\) | = | \((-a+4)^{4}\cdot(-2a+7)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 784 \) | = | \(2^{4}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | $j$ | = | \( \frac{432}{7} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | $r$ | = | \(2\) |
Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.0227095969505290132550084317128574114 \) |
Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 8.0908383878021160530200337268514296456 \) |
Global period: | $\Omega(E/K)$ | ≈ | \( 10.545174114907622496997986618182212107 \) |
Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\) |
Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.8503177441011686065450864444986300260 \) |
Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$\displaystyle 2.850317744 \approx L^{(2)}(E/K,1)/2! \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 10.545174 \cdot 8.090838 \cdot 4 } { {4^2 \cdot 7.483315} } \approx 2.850317744$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.
$\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
---|---|---|---|---|---|---|---|---|
\((-a+4)\) | \(2\) | \(2\) | \(III\) | Additive | \(-1\) | \(3\) | \(4\) | \(0\) |
\((-2a+7)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
56.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 392.d4 |
\(\Q\) | 448.e4 |