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Properties

Label 2.2.5.1-121.1-a2
Base field Q(5)\Q(\sqrt{5})
Conductor norm 121 121
CM no
Base change yes
Q-curve yes
Torsion order 5 5
Rank 0 0

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Base field Q(5)\Q(\sqrt{5})

Generator ϕ\phi, 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+y=x3−x2−10x−20{y}^2+{y}={x}^{3}-{x}^{2}-10{x}-20
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([-10,0]),K([-20,0])])
 
Copy content gp:E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-10,0]),Polrev([-20,0])], K);
 
Copy content magma:E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![-10,0],K![-20,0]]);
 

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/5Z\Z/{5}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(5:−6:1)\left(5 : -6 : 1\right)0055

Invariants

Conductor: N\frak{N} = (11)(11) = (−3ϕ+2)⋅(−3ϕ+1)(-3\phi+2)\cdot(-3\phi+1)
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}) = 121 121 = 11⋅1111\cdot11
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 = −161051-161051
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (−161051)(-161051) = (−3ϕ+2)5⋅(−3ϕ+1)5(-3\phi+2)^{5}\cdot(-3\phi+1)^{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) = 25937424601 25937424601 = 115⋅11511^{5}\cdot11^{5}
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 = −122023936161051 -\frac{122023936}{161051}
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)≈ 1.6108922580697880236230571558378844726 1.6108922580697880236230571558378844726
Tamagawa product: ∏pcp\prod_{\frak{p}}c_{\frak{p}}= 25 25  =  5⋅55\cdot5
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 55
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! ≈0.72041291869443603668204504465720512379 0.72041291869443603668204504465720512379
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

0.720412919≈L(E/K,1)=?#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp#E(K)tor2⋅∣dK∣1/2≈1⋅1.610892⋅1⋅2552⋅2.236068≈0.720412919\begin{aligned}0.720412919 \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 1.610892 \cdot 1 \cdot 25 } { {5^2 \cdot 2.236068} } \\ & \approx 0.720412919 \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))
(−3ϕ+2)(-3\phi+2) 1111 55 I5I_{5} Split multiplicative −1-1 11 55 55
(−3ϕ+1)(-3\phi+1) 1111 55 I5I_{5} Split multiplicative −1-1 11 55 55

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
55 5Cs.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 5.
Its isogeny class 121.1-a consists of curves linked by isogenies of degrees dividing 25.

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 11.a2
Q\Q 275.b2