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Base field \(\Q(\sqrt{-11}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
 
Copy content gp:K = nfinit(Polrev([3, -1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx([3, -1, 1]))
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2638{x}+51031\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-2638,0]),K([51031,0])])
 
Copy content gp:E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,0]),Polrev([-2638,0]),Polrev([51031,0])], K);
 
Copy content magma:E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![-2638,0],K![51031,0]]);
 
Copy content oscar:E = elliptic_curve([K([1,0]),K([1,0]),K([1,0]),K([-2638,0]),K([51031,0])])
 

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

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(\frac{2359}{81} : \frac{1958}{729} a - \frac{11959}{729} : 1\right)$$5.0778424723500213322995616084550874012$$\infty$
$\left(\frac{115}{4} : -\frac{119}{8} : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((-100a+50)\) = \((2)\cdot(-a-1)^{2}\cdot(a-2)^{2}\cdot(-2a+1)\)
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: $N(\frak{N})$ = \( 27500 \) = \(4\cdot5^{2}\cdot5^{2}\cdot11\)
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: $\Delta$ = $945312500$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((945312500)\) = \((2)^{2}\cdot(-a-1)^{9}\cdot(a-2)^{9}\cdot(-2a+1)^{4}\)
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: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 893615722656250000 \) = \(4^{2}\cdot5^{9}\cdot5^{9}\cdot11^{4}\)
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: $j$ = \( \frac{1039509197}{484} \)
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: $r_{\mathrm{an}}$= \( 1 \)
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: $r$ = \(1\)
Regulator: $\mathrm{Reg}(E/K)$ ≈ \( 5.0778424723500213322995616084550874012 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ ≈ \( 10.15568494470004266459912321691017480 \)
Global period: $\Omega(E/K)$≈ \( 0.5148959935235613987043694120896818616 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 32 \)  =  \(2\cdot2\cdot2\cdot2^{2}\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(2\)
Special value: $L^{(r)}(E/K,1)/r!$ ≈\( 12.613115610109796583390825583095148207 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}12.613115610 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.514896 \cdot 10.155685 \cdot 32 } { {2^2 \cdot 3.316625} } \\ & \approx 12.613115610 \end{aligned}$$

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 not semistable. There are 4 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((2)\) \(4\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-a-1)\) \(5\) \(2\) \(III^{*}\) Additive \(-1\) \(2\) \(9\) \(0\)
\((a-2)\) \(5\) \(2\) \(III^{*}\) Additive \(-1\) \(2\) \(9\) \(0\)
\((-2a+1)\) \(11\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 27500.3-t 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
\(\Q\) 550.k1
\(\Q\) 6050.q1