Properties

Label 2.0.3.1-6561.1-CMc1
Base field Q(3)\Q(\sqrt{-3})
Conductor norm 6561 6561
CM yes (3-3)
Base change no
Q-curve yes
Torsion order 3 3
Rank 0 0

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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+(a+1)y=x3{y}^2+\left(a+1\right){y}={x}^{3}
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([1,1]),K([0,0]),K([0,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([1,1]),Polrev([0,0]),Polrev([0,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![1,1],K![0,0],K![0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

Z/3Z\Z/{3}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(0:a1:1)\left(0 : -a - 1 : 1\right)0033

Invariants

Conductor: N\frak{N} = (81)(81) = (2a+1)8(-2a+1)^{8}
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 6561 6561 = 383^{8}
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 243a+243-243a+243
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (243a+243)(-243a+243) = (2a+1)10(-2a+1)^{10}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 59049 59049 = 3103^{10}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 0 0
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z[(1+3)/2]\Z[(1+\sqrt{-3})/2]    (complex multiplication)
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}}) = Z[(1+3)/2]\Z[(1+\sqrt{-3})/2]   
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = U(1)\mathrm{U}(1)

BSD invariants

Analytic rank: ranr_{\mathrm{an}}= 0 0
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: rr = 00
Regulator: Reg(E/K)\mathrm{Reg}(E/K) = 1 1
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) = 1 1
Global period: Ω(E/K)\Omega(E/K) 11.244417652812169700208503930265098418 11.244417652812169700208503930265098418
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 3 3
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 33
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 2.1639891862438953589212350211699331258 2.1639891862438953589212350211699331258
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

2.163989186L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/2111.24441813321.7320512.163989186\displaystyle 2.163989186 \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 11.244418 \cdot 1 \cdot 3 } { {3^2 \cdot 1.732051} } \approx 2.163989186

Local data at primes of bad reduction

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

This elliptic curve is not semistable. There is only one prime 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 33 IVIV Additive 1-1 88 1010 00

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]

For all other primes pp, the image is a split Cartan subgroup if (3p)=+1\left(\frac{ -3 }{p}\right)=+1 or a nonsplit Cartan subgroup if (3p)=1\left(\frac{ -3 }{p}\right)=-1.

Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 6561.1-CMc consists of this curve only.

Base change

This elliptic curve is a Q\Q-curve.

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