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Base field \(\Q(\sqrt{-2}) \)

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

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
 
Copy content gp:K = nfinit(Polrev([2, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx([2, 0, 1]))
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+{x}^{2}-6576{x}-247536\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([-6576,0]),K([-247536,0])])
 
Copy content gp:E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,0]),Polrev([-6576,0]),Polrev([-247536,0])], K);
 
Copy content magma:E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![-6576,0],K![-247536,0]]);
 
Copy content oscar:E = elliptic_curve([K([0,1]),K([1,0]),K([0,0]),K([-6576,0]),K([-247536,0])])
 

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

\(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-\frac{1857}{32} : -\frac{36789}{256} a : 1\right)$$1.9134908847442385566857514407286902998$$\infty$
$\left(12 a - 48 : 24 a + 12 : 1\right)$$0$$2$
$\left(-49 : -120 a : 1\right)$$0$$4$

Invariants

Conductor: $\frak{N}$ = \((204)\) = \((a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\)
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: $N(\frak{N})$ = \( 41616 \) = \(2^{4}\cdot3\cdot3\cdot17\cdot17\)
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: $\Delta$ = $-8036072572032$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-8036072572032)\) = \((a)^{14}\cdot(-a-1)^{2}\cdot(a-1)^{2}\cdot(-2a+3)^{8}\cdot(2a+3)^{8}\)
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: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 64578462382965003828609024 \) = \(2^{14}\cdot3^{2}\cdot3^{2}\cdot17^{8}\cdot17^{8}\)
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: $j$ = \( -\frac{491411892194497}{125563633938} \)
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: $\mathrm{Reg}(E/K)$ ≈ \( 1.9134908847442385566857514407286902998 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ ≈ \( 3.826981769488477113371502881457380600 \)
Global period: $\Omega(E/K)$≈ \( 0.1837541472119480861401817966266871005 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 1024 \)  =  \(2^{2}\cdot2\cdot2\cdot2^{3}\cdot2^{3}\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(8\)
Special value: $L^{(r)}(E/K,1)/r!$ ≈\( 3.9780343798598310070341526577267025247 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}3.978034380 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.183754 \cdot 3.826982 \cdot 1024 } { {8^2 \cdot 2.828427} } \\ & \approx 3.978034380 \end{aligned}$$

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 5 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((a)\) \(2\) \(4\) \(I_{6}^{*}\) Additive \(1\) \(4\) \(14\) \(2\)
\((-a-1)\) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a-1)\) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-2a+3)\) \(17\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((2a+3)\) \(17\) \(8\) \(I_{8}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 41616.5-g 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
\(\Q\) 816.b3
\(\Q\) 3264.m3