Properties

Label 2.0.3.1-138384.2-b2
Base field Q(3)\Q(\sqrt{-3})
Conductor norm 138384 138384
CM no
Base change no
Q-curve yes
Torsion order 3 3
Rank 2 2

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Base field Q(3)\Q(\sqrt{-3})

Generator aa, with minimal polynomial x2x+1 x^{2} - x + 1 ; class number 11.

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

Weierstrass equation

y2=x3+(36a+3)x+84a47{y}^2={x}^{3}+\left(-36a+3\right){x}+84a-47
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([3,-36]),K([-47,84])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([3,-36]),Polrev([-47,84])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![3,-36],K![-47,84]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

ZZZ/3Z\Z \oplus \Z \oplus \Z/{3}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(4a+1:6a5:1)\left(4 a + 1 : 6 a - 5 : 1\right)1.18683647843583816979132127213693498921.1868364784358381697913212721369349892\infty
(a+1:9a:1)\left(-a + 1 : -9 a : 1\right)1.25108429758118343875408656574974748711.2510842975811834387540865657497474871\infty
(3:6a5:1)\left(3 : 6 a - 5 : 1\right)0033

Invariants

Conductor: N\frak{N} = (372)(372) = (2a+1)2(2)2(6a+1)(6a5)(-2a+1)^{2}\cdot(2)^{2}\cdot(-6a+1)\cdot(6a-5)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 138384 138384 = 324231313^{2}\cdot4^{2}\cdot31\cdot31
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 321408a147312-321408a-147312
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (321408a147312)(-321408a-147312) = (2a+1)6(2)4(6a+1)(6a5)3(-2a+1)^{6}\cdot(2)^{4}\cdot(-6a+1)\cdot(6a-5)^{3}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 172351183104 172351183104 = 3644313133^{6}\cdot4^{4}\cdot31\cdot31^{3}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 18504806429791a+37946188829791 -\frac{185048064}{29791} a + \frac{379461888}{29791}
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z   
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}}) = Z\Z    (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)

BSD invariants

Analytic rank: ranr_{\mathrm{an}}= 2 2
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: rr = 22
Regulator: Reg(E/K)\mathrm{Reg}(E/K) 0.46383067915812151697601387049061572289 0.46383067915812151697601387049061572289
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 1.85532271663248606790405548196246289156 1.85532271663248606790405548196246289156
Global period: Ω(E/K)\Omega(E/K) 2.8798055469335009862035990878990204366 2.8798055469335009862035990878990204366
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 18 18  =  23132\cdot3\cdot1\cdot3
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 33
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 6.1695287775182601686499565680088645650 6.1695287775182601686499565680088645650
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

6.169528778L(2)(E/K,1)/2!=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/212.8798061.85532318321.7320516.169528778\displaystyle 6.169528778 \approx L^{(2)}(E/K,1)/2! \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 2.879806 \cdot 1.855323 \cdot 18 } { {3^2 \cdot 1.732051} } \approx 6.169528778

Local data at primes of bad reduction

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

This elliptic curve is not semistable. There are 4 primes p\frak{p} of bad reduction.

p\mathfrak{p} N(p)N(\mathfrak{p}) Tamagawa number Kodaira symbol Reduction type Root number ordp(N\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}) ordp(Dmin\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}) ordp(den(j))\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))
(2a+1)(-2a+1) 33 22 I0I_0^{*} Additive 1-1 22 66 00
(2)(2) 44 33 IVIV Additive 11 22 44 00
(6a+1)(-6a+1) 3131 11 I1I_{1} Split multiplicative 1-1 11 11 11
(6a5)(6a-5) 3131 33 I3I_{3} Split multiplicative 1-1 11 33 33

Galois Representations

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

prime Image of Galois Representation
33 3B.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 3.
Its isogeny class 138384.2-b consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is a Q\Q-curve.

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