Properties

Label 2.0.4.1-32000.2-f1
Base field Q(1)\Q(\sqrt{-1})
Conductor norm 32000 32000
CM no
Base change no
Q-curve no
Torsion order 2 2
Rank 1 1

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Base field Q(1)\Q(\sqrt{-1})

Generator ii, with minimal polynomial x2+1 x^{2} + 1 ; class number 11.

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

Weierstrass equation

y2=x3+x2+(532i534)x+7530i3252{y}^2={x}^{3}+{x}^{2}+\left(532i-534\right){x}+7530i-3252
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-534,532]),K([-3252,7530])])
 
Copy content gp:E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([0,0]),Polrev([-534,532]),Polrev([-3252,7530])], K);
 
Copy content magma:E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-534,532],K![-3252,7530]]);
 

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

ZZ/2Z\Z \oplus \Z/{2}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(2i18:25i+25:1)\left(2 i - 18 : -25 i + 25 : 1\right)0.812534885260938295405915444897960093460.81253488526093829540591544489796009346\infty
(6i15:0:1)\left(6 i - 15 : 0 : 1\right)0022

Invariants

Conductor: N\frak{N} = (80i160)(-80i-160) = (i+1)8(i2)2(2i+1)(i+1)^{8}\cdot(-i-2)^{2}\cdot(2i+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}) = 32000 32000 = 285252^{8}\cdot5^{2}\cdot5
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 = 793600i+435200793600i+435200
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (793600i+435200)(793600i+435200) = (i+1)21(i2)6(2i+1)2(i+1)^{21}\cdot(-i-2)^{6}\cdot(2i+1)^{2}
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) = 819200000000 819200000000 = 22156522^{21}\cdot5^{6}\cdot5^{2}
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 = 35840001425i125950080225 \frac{358400014}{25} i - \frac{1259500802}{25}
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}}= 1 1
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: rr = 11
Regulator: Reg(E/K)\mathrm{Reg}(E/K) 0.81253488526093829540591544489796009346 0.81253488526093829540591544489796009346
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 1.62506977052187659081183088979592018692 1.62506977052187659081183088979592018692
Global period: Ω(E/K)\Omega(E/K) 1.17404614738686914652730130763337763422 1.17404614738686914652730130763337763422
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 16 16  =  22222^{2}\cdot2\cdot2
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 22
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 3.8158138066321454916424828883538092399 3.8158138066321454916424828883538092399
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

3.815813807L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/211.1740461.62507016222.0000003.815813807\begin{aligned}3.815813807 \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 1.174046 \cdot 1.625070 \cdot 16 } { {2^2 \cdot 2.000000} } \\ & \approx 3.815813807 \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 are 3 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))
(i+1)(i+1) 22 44 I9I_{9}^{*} Additive 1-1 88 2121 00
(i2)(-i-2) 55 22 I0I_0^{*} Additive 11 22 66 00
(2i+1)(2i+1) 55 22 I2I_{2} Split multiplicative 1-1 11 22 22

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 4 and 8.
Its isogeny class 32000.2-f consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is not a Q\Q-curve.

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