Properties

Label 2.2.56.1-56.1-d3
Base field \(\Q(\sqrt{14}) \)
Conductor norm \( 56 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 2 \)

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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

\({y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-15{x}-5\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-15,0]),K([-5,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,1]),Polrev([-15,0]),Polrev([-5,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![-15,0],K![-5,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z \oplus \Z \oplus \Z/{4}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(a - 2 : a - 7 : 1\right)$$0.74026689492463658723488979844289176126$$\infty$
$\left(-a - 2 : 7 : 1\right)$$0.74026689492463658723488979844289176126$$\infty$
$\left(5 : -3 a - 14 : 1\right)$$0$$4$

Invariants

Conductor: $\frak{N}$ = \((2a)\) = \((-a+4)^{3}\cdot(-2a+7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 56 \) = \(2^{3}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $76832$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((76832)\) = \((-a+4)^{10}\cdot(-2a+7)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 5903156224 \) = \(2^{10}\cdot7^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( \frac{11090466}{2401} \)
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: $r_{\mathrm{an}}$= \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: $r$ = \(2\)
Regulator: $\mathrm{Reg}(E/K)$ \( 0.50567739923763225331375210792821435285 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 2.02270959695052901325500843171285741140 \)
Global period: $\Omega(E/K)$ \( 10.545174114907622496997986618182212107 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 16 \)  =  \(2\cdot2^{3}\)
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

$\displaystyle 2.850317744 \approx L^{(2)}(E/K,1)/2! \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 10.545174 \cdot 2.022710 \cdot 16 } { {4^2 \cdot 7.483315} } \approx 2.850317744$

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 \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-a+4)\) \(2\) \(2\) \(III^{*}\) Additive \(-1\) \(3\) \(10\) \(0\)
\((-2a+7)\) \(7\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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.d2
\(\Q\) 448.e2