Properties

Label 2.2.33.1-539.1-a1
Base field Q(33)\Q(\sqrt{33})
Conductor norm 539 539
CM no
Base change yes
Q-curve yes
Torsion order 1 1
Rank 0 0

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Base field Q(33)\Q(\sqrt{33})

Generator aa, with minimal polynomial x2x8 x^{2} - x - 8 ; class number 11.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, -1, 1]))
 
gp: K = nfinit(Polrev([-8, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);
 

Weierstrass equation

y2+y=x3ax2+(32875a110860)x+5524839a18631313{y}^2+{y}={x}^{3}-a{x}^{2}+\left(32875a-110860\right){x}+5524839a-18631313
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([1,0]),K([-110860,32875]),K([-18631313,5524839])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([1,0]),Polrev([-110860,32875]),Polrev([-18631313,5524839])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![1,0],K![-110860,32875],K![-18631313,5524839]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

trivial

Invariants

Conductor: N\frak{N} = (28a63)(-28a-63) = (4a9)(7)(-4a-9)\cdot(7)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 539 539 = 114911\cdot49
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 539-539
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (539)(-539) = (4a9)2(7)2(-4a-9)^{2}\cdot(7)^{2}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 290521 290521 = 11249211^{2}\cdot49^{2}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 78843215872539 -\frac{78843215872}{539}
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}}= 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) 0.60288154826947420090237316500130575728 0.60288154826947420090237316500130575728
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 4 4  =  222\cdot2
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 11
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 1.6791713078670715093471802716613307997 1.6791713078670715093471802716613307997
Analytic order of Ш: Шan{}_{\mathrm{an}}= 4 4 (rounded)

BSD formula

1.679171308L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/240.60288214125.7445631.679171308\displaystyle 1.679171308 \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{ 4 \cdot 0.602882 \cdot 1 \cdot 4 } { {1^2 \cdot 5.744563} } \approx 1.679171308

Local data at primes of bad reduction

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

This elliptic curve is semistable. There are 2 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))
(4a9)(-4a-9) 1111 22 I2I_{2} Split multiplicative 1-1 11 22 22
(7)(7) 4949 22 I2I_{2} Split multiplicative 1-1 11 22 22

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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 3 and 9.
Its isogeny class 539.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

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

Base field Curve
Q\Q 693.b1
Q\Q 847.c1