Properties

Label 2.2.5.1-2401.1-b3
Base field Q(5)\Q(\sqrt{5})
Conductor norm 2401 2401
CM yes (35-35)
Base change no
Q-curve yes
Torsion order 1 1
Rank 0 0

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

Generator ϕ\phi, with minimal polynomial x2x1 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=x3x2+(14ϕ+5)x21ϕ15{y}^2+{y}={x}^{3}-{x}^{2}+\left(14\phi+5\right){x}-21\phi-15
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([5,14]),K([-15,-21])])
 
Copy content gp:E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([5,14]),Polrev([-15,-21])], K);
 
Copy content magma:E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![5,14],K![-15,-21]]);
 

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

trivial

Invariants

Conductor: N\frak{N} = (49)(49) = (7)2(7)^{2}
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}) = 2401 2401 = 49249^{2}
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 = 343-343
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (343)(-343) = (7)3(7)^{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) = 117649 117649 = 49349^{3}
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 = 52756480ϕ85360640 52756480 \phi - 85360640
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[(1+35)/2]\Z[(1+\sqrt{-35})/2]    (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) = N(U(1))N(\mathrm{U}(1))

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) 2.4377555092472808403399055475038616336 2.4377555092472808403399055475038616336
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 2 2
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 11
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 2.1803948124806148589184806540746934821 2.1803948124806148589184806540746934821
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

2.180394812L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/212.43775612122.2360682.180394812\begin{aligned}2.180394812 \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 2.437756 \cdot 1 \cdot 2 } { {1^2 \cdot 2.236068} } \\ & \approx 2.180394812 \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 not semistable. There is only one prime 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))
(7)(7) 4949 22 IIIIII Additive 11 22 33 00

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.1.5

For all other primes pp, the image is a Borel subgroup if p=5p=5, the normalizer of a split Cartan subgroup if (35p)=+1\left(\frac{ -35 }{p}\right)=+1 or the normalizer of a nonsplit Cartan subgroup if (35p)=1\left(\frac{ -35 }{p}\right)=-1.

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 5, 7 and 35.
Its isogeny class 2401.1-b consists of curves linked by isogenies of degrees dividing 35.

Base change

This elliptic curve is a Q\Q-curve.

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