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Properties

Label 2.2.21.1-28.1-a3
Base field Q(21)\Q(\sqrt{21})
Conductor norm 28 28
CM no
Base change yes
Q-curve yes
Torsion order 6 6
Rank 0 0

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Base field Q(21)\Q(\sqrt{21})

Generator aa, with minimal polynomial x2−x−5 x^{2} - x - 5 ; class number 11.

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -1, 1]))
 
Copy content gp:K = nfinit(Polrev([-5, -1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

y2+axy+(a+1)y=x3+x2+(−23a+62)x−161a+447{y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-23a+62\right){x}-161a+447
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([1,1]),K([62,-23]),K([447,-161])])
 
Copy content gp:E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([1,1]),Polrev([62,-23]),Polrev([447,-161])], K);
 
Copy content magma:E := EllipticCurve([K![0,1],K![1,0],K![1,1],K![62,-23],K![447,-161]]);
 

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/6Z\Z/{6}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(−5a+14:23a−65:1)\left(-5 a + 14 : 23 a - 65 : 1\right)0066

Invariants

Conductor: N\frak{N} = (2a+6)(2a+6) = (2)⋅(a+3)(2)\cdot(a+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}) = 28 28 = 4⋅74\cdot7
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 = −21952-21952
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (−21952)(-21952) = (2)6⋅(a+3)6(2)^{6}\cdot(a+3)^{6}
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) = 481890304 481890304 = 46⋅764^{6}\cdot7^{6}
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 = 993837521952 \frac{9938375}{21952}
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)≈ 7.0277081059000617705423007125374356141 7.0277081059000617705423007125374356141
Tamagawa product: ∏pcp\prod_{\frak{p}}c_{\frak{p}}= 36 36  =  (2⋅3)⋅(2⋅3)( 2 \cdot 3 )\cdot( 2 \cdot 3 )
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 66
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! ≈1.5335716360638936011918161977277412414 1.5335716360638936011918161977277412414
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

1.533571636≈L(E/K,1)=?#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp#E(K)tor2⋅∣dK∣1/2≈1⋅7.027708⋅1⋅3662⋅4.582576≈1.533571636\begin{aligned}1.533571636 \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 7.027708 \cdot 1 \cdot 36 } { {6^2 \cdot 4.582576} } \\ & \approx 1.533571636 \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) 44 66 I6I_{6} Split multiplicative −1-1 11 66 66
(a+3)(a+3) 77 66 I6I_{6} Split multiplicative −1-1 11 66 66

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
22 2B
33 3Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 3 and 6.
Its isogeny class 28.1-a consists of curves linked by isogenies of degrees dividing 18.

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 98.a6
Q\Q 126.b6