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Properties

Label 2.0.8.1-17424.5-b6
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 17424 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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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]);
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+{x}^{2}+460{x}+4488\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([460,0]),K([4488,0])])
 
Copy content gp:E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,0]),Polrev([460,0]),Polrev([4488,0])], K);
 
Copy content magma:E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![460,0],K![4488,0]]);
 

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

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-16 a - 21 : 73 a - 176 : 1\right)$$1.8137303950654075072686213898323383261$$\infty$
$\left(-\frac{17}{2} : \frac{17}{4} a : 1\right)$$0$$2$
$\left(-16 a + 4 : -2 a - 16 : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((132)\) = \((a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(a+3)\cdot(a-3)\)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 17424 \) = \(2^{4}\cdot3\cdot3\cdot11\cdot11\)
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Discriminant: $\Delta$ = $-14632814592$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-14632814592)\) = \((a)^{22}\cdot(-a-1)^{10}\cdot(a-1)^{10}\cdot(a+3)^{2}\cdot(a-3)^{2}\)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 214119262883848126464 \) = \(2^{22}\cdot3^{10}\cdot3^{10}\cdot11^{2}\cdot11^{2}\)
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));
 
j-invariant: $j$ = \( \frac{168105213359}{228637728} \)
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(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: $\mathrm{Reg}(E/K)$ ≈ \( 1.8137303950654075072686213898323383261 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ ≈ \( 3.6274607901308150145372427796646766522 \)
Global period: $\Omega(E/K)$≈ \( 0.560225554238699981274849680716370886460 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 64 \)  =  \(2^{2}\cdot2\cdot2\cdot2\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(4\)
Special value: $L^{(r)}(E/K,1)/r!$ ≈\( 2.8739594721749081514005950388742900729 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}2.873959472 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.560226 \cdot 3.627461 \cdot 64 } { {4^2 \cdot 2.828427} } \\ & \approx 2.873959472 \end{aligned}$$

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 \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((a)\) \(2\) \(4\) \(I_{14}^{*}\) Additive \(1\) \(4\) \(22\) \(10\)
\((-a-1)\) \(3\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\((a-1)\) \(3\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\((a+3)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a-3)\) \(11\) \(2\) \(I_{2}\) Non-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
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 17424.5-b consists of curves linked by isogenies of degrees dividing 20.

Base change

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

Base field Curve
\(\Q\) 528.a4
\(\Q\) 2112.n4