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Properties

Label 3.3.49.1-216.1-b2
Base field Q(ζ7)+\Q(\zeta_{7})^+
Conductor norm 216 216
CM no
Base change yes
Q-curve yes
Torsion order 7 7
Rank 0 0

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Base field Q(ζ7)+\Q(\zeta_{7})^+

Generator aa, with minimal polynomial x3−x2−2x+1 x^{3} - x^{2} - 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, -2, -1, 1]))
 
Copy content gp:K = nfinit(Polrev([1, -2, -1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
 

Weierstrass equation

y2+(a+1)xy+(a2+a−2)y=x3+(−a2−a+2)x2+(−a2−a+1)x−a2+1{y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-a^{2}-a+1\right){x}-a^{2}+1
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,1,0]),K([2,-1,-1]),K([-2,1,1]),K([1,-1,-1]),K([1,0,-1])])
 
Copy content gp:E = ellinit([Polrev([1,1,0]),Polrev([2,-1,-1]),Polrev([-2,1,1]),Polrev([1,-1,-1]),Polrev([1,0,-1])], K);
 
Copy content magma:E := EllipticCurve([K![1,1,0],K![2,-1,-1],K![-2,1,1],K![1,-1,-1],K![1,0,-1]]);
 

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/7Z\Z/{7}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(0:−a:1)\left(0 : -a : 1\right)0077

Invariants

Conductor: N\frak{N} = (6)(6) = (2)⋅(3)(2)\cdot(3)
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}) = 216 216 = 8⋅278\cdot27
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 = −6-6
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (−6)(-6) = (2)⋅(3)(2)\cdot(3)
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) = −216 -216 = −8⋅27-8\cdot27
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 = −24016 -\frac{2401}{6}
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)≈ 299.31257229538945214023709328525443451 299.31257229538945214023709328525443451
Tamagawa product: ∏pcp\prod_{\frak{p}}c_{\frak{p}}= 1 1  =  1⋅11\cdot1
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 77
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! ≈0.87263140610900714909690114660424033384 0.87263140610900714909690114660424033384
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

0.872631406≈L(E/K,1)=?#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp#E(K)tor2⋅∣dK∣1/2≈1⋅299.312572⋅1⋅172⋅7.000000≈0.872631406\begin{aligned}0.872631406 \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 299.312572 \cdot 1 \cdot 1 } { {7^2 \cdot 7.000000} } \\ & \approx 0.872631406 \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 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))
(2)(2) 88 11 I1I_{1} Split multiplicative −1-1 11 11 11
(3)(3) 2727 11 I1I_{1} Non-split multiplicative 11 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
77 7B.1.1[3]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 7.
Its isogeny class 216.1-b consists of curves linked by isogenies of degree 7.

Base change

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

Base field Curve
Q\Q 294.e2