Properties

Label 2.2.56.1-56.1-c1
Base field \(\Q(\sqrt{14}) \)
Conductor norm \( 56 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{14}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))
 
gp: K = nfinit(Polrev([-14, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(7a-33\right){x}+1856a-6948\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([-33,7]),K([-6948,1856])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,1]),Polrev([-33,7]),Polrev([-6948,1856])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,1],K![-33,7],K![-6948,1856]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-19 a + 71 : -268 a + 1001 : 1\right)$$0.84449381520494372590220177553615918844$$\infty$
$\left(-3 a + 11 : -6 a + 21 : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((2a)\) = \((-a+4)^{3}\cdot(-2a+7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 56 \) = \(2^{3}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $-112$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-112)\) = \((-a+4)^{8}\cdot(-2a+7)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 12544 \) = \(2^{8}\cdot7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( -\frac{4}{7} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
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);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: $r$ = \(1\)
Regulator: $\mathrm{Reg}(E/K)$ \( 0.84449381520494372590220177553615918844 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 1.68898763040988745180440355107231837688 \)
Global period: $\Omega(E/K)$ \( 7.1899219485544091578329242905522858392 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 4 \)  =  \(2\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(2\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 1.6227687331350300805033443351553630957 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 1.622768733 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 7.189922 \cdot 1.688988 \cdot 4 } { {2^2 \cdot 7.483315} } \approx 1.622768733$

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 

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 \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-a+4)\) \(2\) \(2\) \(I_{1}^{*}\) Additive \(1\) \(3\) \(8\) \(0\)
\((-2a+7)\) \(7\) \(2\) \(I_{2}\) 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.
Its isogeny class 56.1-c consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 56.b2
\(\Q\) 3136.c2