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Properties

Label 4.4.2048.1-1.1-a10
Base field Q(ζ16)+\Q(\zeta_{16})^+
Conductor norm 1 1
CM no
Base change no
Q-curve yes
Torsion order 2 2
Rank 0 0

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Base field Q(ζ16)+\Q(\zeta_{16})^+

Generator aa, with minimal polynomial x4−4x2+2 x^{4} - 4 x^{2} + 2 ; class number 11.

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -4, 0, 1]))
 
Copy content gp:K = nfinit(Polrev([2, 0, -4, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -4, 0, 1]);
 

Weierstrass equation

y2+(a3−3a)xy+y=x3+(a3−a2−3a+2)x2+(−113a3+112a2+423a−455)x−1225a3+1100a2+4334a−4038{y}^2+\left(a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-113a^{3}+112a^{2}+423a-455\right){x}-1225a^{3}+1100a^{2}+4334a-4038
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,-3,0,1]),K([2,-3,-1,1]),K([1,0,0,0]),K([-455,423,112,-113]),K([-4038,4334,1100,-1225])])
 
Copy content gp:E = ellinit([Polrev([0,-3,0,1]),Polrev([2,-3,-1,1]),Polrev([1,0,0,0]),Polrev([-455,423,112,-113]),Polrev([-4038,4334,1100,-1225])], K);
 
Copy content magma:E := EllipticCurve([K![0,-3,0,1],K![2,-3,-1,1],K![1,0,0,0],K![-455,423,112,-113],K![-4038,4334,1100,-1225]]);
 

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/2Z\Z/{2}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(2a3−194a2−11a+23:−738a3+72a2+1194a−192:1)\left(2 a^{3} - \frac{19}{4} a^{2} - 11 a + 23 : -\frac{73}{8} a^{3} + \frac{7}{2} a^{2} + \frac{119}{4} a - \frac{19}{2} : 1\right)0022

Invariants

Conductor: N\frak{N} = (1)(1) = (1)(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}) = 1 1 = 1
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 = 11
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (1)(1) = (1)(1)
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) = 1 1 = 1
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 = 4374624644505560827923496904a3+8083252342956303105729121856a2−2562595786459777953350768848a−4735059594419705758581752312 4374624644505560827923496904 a^{3} + 8083252342956303105729121856 a^{2} - 2562595786459777953350768848 a - 4735059594419705758581752312
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)≈ 1.6302687553063589652205890031213109992 1.6302687553063589652205890031213109992
Tamagawa product: ∏pcp\prod_{\frak{p}}c_{\frak{p}}= 1 1
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 22
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! ≈0.225151189850328 0.225151189850328
Analytic order of Ш: Шan{}_{\mathrm{an}}= 25 25 (rounded)

BSD formula

0.225151190≈L(E/K,1)=?#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp#E(K)tor2⋅∣dK∣1/2≈25⋅1.630269⋅1⋅122⋅45.254834≈0.225151190\begin{aligned}0.225151190 \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{ 25 \cdot 1.630269 \cdot 1 \cdot 1 } { {2^2 \cdot 45.254834} } \\ & \approx 0.225151190 \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 no primes of bad reduction.

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
55 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 4, 5, 8, 10, 20 and 40.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 40.

Base change

This elliptic curve is a Q\Q-curve.

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