Properties

Label 6.6.434581.1-71.1-b1
Base field 6.6.434581.1
Conductor norm \( 71 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.434581.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 4, 5, -4, -2, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 4, 5, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(3a^{5}-6a^{4}-10a^{3}+12a^{2}+4a-2\right){x}{y}+\left(a^{5}-2a^{4}-3a^{3}+4a^{2}-1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}+4a^{2}-5a-3\right){x}^{2}+\left(-297a^{5}+798a^{4}+604a^{3}-1840a^{2}+208a+366\right){x}-3433a^{5}+9442a^{4}+6587a^{3}-22010a^{2}+3003a+4476\)
sage: E = EllipticCurve([K([-2,4,12,-10,-6,3]),K([-3,-5,4,6,0,-1]),K([-1,0,4,-3,-2,1]),K([366,208,-1840,604,798,-297]),K([4476,3003,-22010,6587,9442,-3433])])
 
gp: E = ellinit([Polrev([-2,4,12,-10,-6,3]),Polrev([-3,-5,4,6,0,-1]),Polrev([-1,0,4,-3,-2,1]),Polrev([366,208,-1840,604,798,-297]),Polrev([4476,3003,-22010,6587,9442,-3433])], K);
 
magma: E := EllipticCurve([K![-2,4,12,-10,-6,3],K![-3,-5,4,6,0,-1],K![-1,0,4,-3,-2,1],K![366,208,-1840,604,798,-297],K![4476,3003,-22010,6587,9442,-3433]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

trivial

Invariants

Conductor: $\frak{N}$ = \((2a^5-6a^4-4a^3+17a^2-a-6)\) = \((2a^5-6a^4-4a^3+17a^2-a-6)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 71 \) = \(71\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $5a^5-14a^4-8a^3+33a^2-6a-16$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((5a^5-14a^4-8a^3+33a^2-6a-16)\) = \((2a^5-6a^4-4a^3+17a^2-a-6)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 357911 \) = \(71^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( -\frac{25415776248234932727}{357911} a^{5} + \frac{33644549177505288315}{357911} a^{4} + \frac{124407707776619205840}{357911} a^{3} - \frac{42935695592915487066}{357911} a^{2} - \frac{130674592083511357736}{357911} a - \frac{37564287471550457576}{357911} \)
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: $r_{\mathrm{an}}$= \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: $r$ = \(0\)
Regulator: $\mathrm{Reg}(E/K)$ = \( 1 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ = \( 1 \)
Global period: $\Omega(E/K)$ \( 9.1228616743419730844511926319767364532 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 1 \)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(1\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 1.12094 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 81 \) (rounded)

BSD formula

$\displaystyle 1.120940000 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 81 \cdot 9.122862 \cdot 1 \cdot 1 } { {1^2 \cdot 659.227578} } \approx 1.120935804$

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 

This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((2a^5-6a^4-4a^3+17a^2-a-6)\) \(71\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 71.1-b consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.