Properties

Label 2.0.4.1-76250.5-a1
Base field Q(1)\Q(\sqrt{-1})
Conductor norm 76250 76250
CM no
Base change no
Q-curve no
Torsion order 1 1
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+xy+iy=x3+(i1)x2+(i+166)x927i367{y}^2+{x}{y}+i{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-i+166\right){x}-927i-367
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,0]),K([-1,1]),K([0,1]),K([166,-1]),K([-367,-927])])
 
Copy content gp:E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([0,1]),Polrev([166,-1]),Polrev([-367,-927])], K);
 
Copy content magma:E := EllipticCurve([K![1,0],K![-1,1],K![0,1],K![166,-1],K![-367,-927]]);
 

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\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(21i19:58i+146:1)\left(-21 i - 19 : -58 i + 146 : 1\right)0.129453496236471816236226991074002634080.12945349623647181623622699107400263408\infty

Invariants

Conductor: N\frak{N} = (25i+275)(25i+275) = (i+1)(i2)2(2i+1)2(6i5)(i+1)\cdot(-i-2)^{2}\cdot(2i+1)^{2}\cdot(-6i-5)
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}) = 76250 76250 = 25252612\cdot5^{2}\cdot5^{2}\cdot61
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 = 276523125i+77556875-276523125i+77556875
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (276523125i+77556875)(-276523125i+77556875) = (i+1)(i2)7(2i+1)4(6i5)5(i+1)\cdot(-i-2)^{7}\cdot(2i+1)^{4}\cdot(-6i-5)^{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) = 82480107519531250 82480107519531250 = 257546152\cdot5^{7}\cdot5^{4}\cdot61^{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 = 143622832666838445963010i40282121360698445963010 -\frac{14362283266683}{8445963010} i - \frac{4028212136069}{8445963010}
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) 0.12945349623647181623622699107400263408 0.12945349623647181623622699107400263408
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 0.258906992472943632472453982148005268160 0.258906992472943632472453982148005268160
Global period: Ω(E/K)\Omega(E/K) 1.04647644809975611835187191133278244792 1.04647644809975611835187191133278244792
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 20 20  =  122151\cdot2^{2}\cdot1\cdot5
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 11
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 2.7094006987127634519724927618884770900 2.7094006987127634519724927618884770900
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

2.709400699L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/211.0464760.25890720122.0000002.709400699\begin{aligned}2.709400699 \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 1.046476 \cdot 0.258907 \cdot 20 } { {1^2 \cdot 2.000000} } \\ & \approx 2.709400699 \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 4 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 11 I1I_{1} Non-split multiplicative 11 11 11 11
(i2)(-i-2) 55 44 I1I_{1}^{*} Additive 11 22 77 11
(2i+1)(2i+1) 55 11 IVIV Additive 1-1 22 44 00
(6i5)(-6i-5) 6161 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 5B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 5.
Its isogeny class 76250.5-a consists of curves linked by isogenies of degree 5.

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.