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Properties

Label 2.0.3.1-868.2-b1
Base field Q(−3)\Q(\sqrt{-3})
Conductor norm 868 868
CM no
Base change no
Q-curve no
Torsion order 3 3
Rank 0 0

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Base field Q(−3)\Q(\sqrt{-3})

Generator aa, with minimal polynomial x2−x+1 x^{2} - x + 1 ; class number 11.

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
 
Copy content gp:K = nfinit(Polrev([1, -1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

y2+xy+(a+1)y=x3+(a+1)x2+(−1457a−824)x−35350a−779{y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1457a-824\right){x}-35350a-779
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,0]),K([1,1]),K([1,1]),K([-824,-1457]),K([-779,-35350])])
 
Copy content gp:E = ellinit([Polrev([1,0]),Polrev([1,1]),Polrev([1,1]),Polrev([-824,-1457]),Polrev([-779,-35350])], K);
 
Copy content magma:E := EllipticCurve([K![1,0],K![1,1],K![1,1],K![-824,-1457],K![-779,-35350]]);
 

This is a global minimal model.

Copy content comment:Test whether it is a global minimal model
 
Copy content sage:E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(−313a−593:349a+889:1)\left(-\frac{31}{3} a - \frac{59}{3} : \frac{34}{9} a + \frac{88}{9} : 1\right)0033

Invariants

Conductor: N\frak{N} = (32a−6)(32a-6) = (2)⋅(−3a+1)⋅(6a−5)(2)\cdot(-3a+1)\cdot(6a-5)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 868 868 = 4⋅7⋅314\cdot7\cdot31
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Discriminant: Δ\Delta = −11264a−11776-11264a-11776
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (−11264a−11776)(-11264a-11776) = (2)9⋅(−3a+1)2⋅(6a−5)(2)^{9}\cdot(-3a+1)^{2}\cdot(6a-5)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 398196736 398196736 = 49⋅72⋅314^{9}\cdot7^{2}\cdot31
Copy content comment:Compute the norm of the discriminant
 
Copy content sage:E.discriminant().norm()
 
Copy content gp:norm(E.disc)
 
Copy content magma:Norm(Discriminant(E));
 
j-invariant: jj = −19926242340409933388864a+18769373204677155777728 -\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728}
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content 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)
Copy content comment:Test for Complex Multiplication
 
Copy content sage:E.has_cm(), E.cm_discriminant()
 
Copy content 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
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content 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)≈ 0.994093830181886225150316385942862955240 0.994093830181886225150316385942862955240
Tamagawa product: ∏pcp\prod_{\frak{p}}c_{\frak{p}}= 18 18  =  32⋅2⋅13^{2}\cdot2\cdot1
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 33
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! ≈1.1478806809105162671199956532121037293 1.1478806809105162671199956532121037293
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

1.147880681≈L(E/K,1)=?#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp#E(K)tor2⋅∣dK∣1/2≈1⋅0.994094⋅1⋅1832⋅1.732051≈1.147880681\begin{aligned}1.147880681 \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 0.994094 \cdot 1 \cdot 18 } { {3^2 \cdot 1.732051} } \\ & \approx 1.147880681 \end{aligned}

Local data at primes of bad reduction

Copy content comment:Compute the local reduction data at primes of bad reduction
 
Copy content sage:E.local_data()
 
Copy content magma:LocalInformation(E);
 

This elliptic curve is semistable. There are 3 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))
(2)(2) 44 99 I9I_{9} Split multiplicative −1-1 11 99 99
(−3a+1)(-3a+1) 77 22 I2I_{2} Split multiplicative −1-1 11 22 22
(6a−5)(6a-5) 3131 11 I1I_{1} Split multiplicative −1-1 11 11 11

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 and 9.
Its isogeny class 868.2-b consists of curves linked by isogenies of degrees dividing 9.

Base change

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

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