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

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

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
 
Copy content gp:K = nfinit(Polrev([2, -1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx([2, -1, 1]))
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}-{x}^{2}-8{x}+44\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([-8,0]),K([44,0])])
 
Copy content gp:E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([-8,0]),Polrev([44,0])], K);
 
Copy content magma:E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-8,0],K![44,0]]);
 
Copy content oscar:E = elliptic_curve([K([1,0]),K([-1,0]),K([0,0]),K([-8,0]),K([44,0])])
 

This is a global minimal model.

Copy content comment:Test whether it is a global minimal model
 
Copy content sage:E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-3 a + 3 : 7 a + 1 : 1\right)$$0.16689364344617958654816546642605483849$$\infty$
$\left(-\frac{7}{4} a - \frac{15}{4} : \frac{49}{8} a - \frac{83}{8} : 1\right)$$0.71193240221216308625862377745826532143$$\infty$
$\left(-4 : 2 : 1\right)$$0$$2$
$\left(-\frac{7}{4} a + \frac{13}{4} : \frac{7}{8} a - \frac{13}{8} : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((-92a+46)\) = \((a)\cdot(-a+1)\cdot(-2a+1)\cdot(-2a+5)\cdot(2a+3)\)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Conductor norm: $N(\frak{N})$ = \( 14812 \) = \(2\cdot2\cdot7\cdot23\cdot23\)
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(K, ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Copy content oscar:norm(conductor(E))
 
Discriminant: $\Delta$ = $-725788$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-725788)\) = \((a)^{2}\cdot(-a+1)^{2}\cdot(-2a+1)^{6}\cdot(-2a+5)^{2}\cdot(2a+3)^{2}\)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 526768220944 \) = \(2^{2}\cdot2^{2}\cdot7^{6}\cdot23^{2}\cdot23^{2}\)
Copy content comment:Compute the norm of the discriminant
 
Copy content sage:E.discriminant().norm()
 
Copy content gp:norm(E.disc)
 
Copy content magma:Norm(Discriminant(E));
 
Copy content oscar:norm(discriminant(E))
 
j-invariant: $j$ = \( -\frac{60698457}{725788} \)
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z\)    (no potential complex multiplication)
Copy content comment:Test for Complex Multiplication
 
Copy content sage:E.has_cm(), E.cm_discriminant()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 2 \)
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: $r$ = \(2\)
Regulator: $\mathrm{Reg}(E/K)$ ≈ \( 0.11185362043689373062969250900784606903 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ ≈ \( 0.4474144817475749225187700360313842761 \)
Global period: $\Omega(E/K)$≈ \( 2.945509658123824030582349312224125546 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 96 \)  =  \(2\cdot2\cdot( 2 \cdot 3 )\cdot2\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(4\)
Special value: $L^{(r)}(E/K,1)/r!$ ≈\( 2.9886339014417845534553139695421385651 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}2.988633901 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2.945510 \cdot 0.447414 \cdot 96 } { {4^2 \cdot 2.645751} } \\ & \approx 2.988633901 \end{aligned}$$

Local data at primes of bad reduction

Copy content comment:Compute the local reduction data at primes of bad reduction
 
Copy content sage:E.local_data()
 
Copy content magma:LocalInformation(E);
 

This elliptic curve is semistable. There are 5 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((a)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-a+1)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-2a+1)\) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((-2a+5)\) \(23\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((2a+3)\) \(23\) \(2\) \(I_{2}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 14812.5-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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

Base field Curve
\(\Q\) 322.a2
\(\Q\) 2254.d2