Properties

Label 2.0.7.1-567.1-a2
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 567 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{-7}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
 
gp: K = nfinit(Polrev([2, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}-{x}^{2}+9{x}\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([9,0]),K([0,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([9,0]),Polrev([0,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![9,0],K![0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-\frac{9}{4} : -\frac{9}{2} a + \frac{27}{8} : 1\right)$$1.3192544115670790874427516950298956331$$\infty$
$\left(0 : 0 : 1\right)$$0$$2$
$\left(-\frac{9}{4} a + \frac{3}{2} : \frac{9}{8} a - \frac{3}{4} : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((-18a+9)\) = \((-2a+1)\cdot(3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 567 \) = \(7\cdot9^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $-45927$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-45927)\) = \((-2a+1)^{2}\cdot(3)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 2109289329 \) = \(7^{2}\cdot9^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( \frac{103823}{63} \)
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: $\mathrm{Reg}(E/K)$ \( 1.3192544115670790874427516950298956331 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 2.6385088231341581748855033900597912662 \)
Global period: $\Omega(E/K)$ \( 4.5977436247313438967261720911547637236 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 8 \)  =  \(2\cdot2^{2}\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(4\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 2.2925788734620641605396400556355936826 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 2.292578873 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 4.597744 \cdot 2.638509 \cdot 8 } { {4^2 \cdot 2.645751} } \approx 2.292578873$

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 \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-2a+1)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((3)\) \(9\) \(4\) \(I_{2}^{*}\) Additive \(1\) \(2\) \(8\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 567.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

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

Base field Curve
\(\Q\) 63.a6
\(\Q\) 441.f6