Properties

Label 6.6.371293.1-53.4-a1
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 53 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\zeta_{13})^+\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-1\right){x}^{2}+\left(a^{5}-a^{4}-3a^{3}+4a^{2}-a\right){x}+a^{5}-a^{4}-3a^{3}+4a^{2}-1\)
sage: E = EllipticCurve([K([1,0,-3,0,1,0]),K([-1,2,4,-4,-1,1]),K([-2,1,1,0,0,0]),K([0,-1,4,-3,-1,1]),K([-1,0,4,-3,-1,1])])
 
gp: E = ellinit([Polrev([1,0,-3,0,1,0]),Polrev([-1,2,4,-4,-1,1]),Polrev([-2,1,1,0,0,0]),Polrev([0,-1,4,-3,-1,1]),Polrev([-1,0,4,-3,-1,1])], K);
 
magma: E := EllipticCurve([K![1,0,-3,0,1,0],K![-1,2,4,-4,-1,1],K![-2,1,1,0,0,0],K![0,-1,4,-3,-1,1],K![-1,0,4,-3,-1,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-5 a^{5} - 3 a^{4} + 22 a^{3} + 13 a^{2} - 14 a - 3 : -28 a^{5} - 12 a^{4} + 118 a^{3} + 60 a^{2} - 66 a - 17 : 1\right)$$0.0012791912557583409003467950630334646415$$\infty$

Invariants

Conductor: $\frak{N}$ = \((a^3-2a-2)\) = \((a^3-2a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 53 \) = \(53\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $2a^5+a^4-5a^3-9a^2-6a+7$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((2a^5+a^4-5a^3-9a^2-6a+7)\) = \((a^3-2a-2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( -148877 \) = \(-53^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( -\frac{43252627634}{148877} a^{5} + \frac{53595700543}{148877} a^{4} + \frac{203435375839}{148877} a^{3} - \frac{221723599564}{148877} a^{2} - \frac{206277232132}{148877} a + \frac{179199457234}{148877} \)
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: $r_{\mathrm{an}}$= \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: $r$ = \(1\)
Regulator: $\mathrm{Reg}(E/K)$ \( 0.0012791912557583409003467950630334646415 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 0.0076751475345500454020807703782007878490000 \)
Global period: $\Omega(E/K)$ \( 60162.441045792087048848477885082004309 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 3 \)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(1\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 2.27340 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 2.273400000 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 60162.441046 \cdot 0.007675 \cdot 3 } { {1^2 \cdot 609.338166} } \approx 2.273395811$

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 \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((a^3-2a-2)\) \(53\) \(3\) \(I_{3}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 53.4-a 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.