Properties

Label 2.0.7.1-26244.2-l3
Base field Q(7)\Q(\sqrt{-7})
Conductor norm 26244 26244
CM no
Base change yes
Q-curve yes
Torsion order 3 3
Rank 1 1

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Base field Q(7)\Q(\sqrt{-7})

Generator aa, with minimal polynomial x2x+2 x^{2} - x + 2 ; class number 11.

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

Weierstrass equation

y2+xy+y=x3x295x697{y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-95{x}-697
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-95,0]),K([-697,0])])
 
Copy content gp:E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-95,0]),Polrev([-697,0])], K);
 
Copy content magma:E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-95,0],K![-697,0]]);
 

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

ZZ/3Z\Z \oplus \Z/{3}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(4a1:12a6:1)\left(-4 a - 1 : -12 a - 6 : 1\right)0.239900864625669043131364530794130080010.23990086462566904313136453079413008001\infty
(19:74:1)\left(19 : -74 : 1\right)0033

Invariants

Conductor: N\frak{N} = (162)(162) = (a)(a+1)(3)4(a)\cdot(-a+1)\cdot(3)^{4}
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}) = 26244 26244 = 22942\cdot2\cdot9^{4}
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 = 169869312-169869312
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (169869312)(-169869312) = (a)21(a+1)21(3)4(a)^{21}\cdot(-a+1)^{21}\cdot(3)^{4}
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) = 28855583159353344 28855583159353344 = 221221942^{21}\cdot2^{21}\cdot9^{4}
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 = 11590886252097152 -\frac{1159088625}{2097152}
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) 0.23990086462566904313136453079413008001 0.23990086462566904313136453079413008001
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 0.479801729251338086262729061588260160020 0.479801729251338086262729061588260160020
Global period: Ω(E/K)\Omega(E/K) 1.16697043968117750273764330319746246122 1.16697043968117750273764330319746246122
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 441 441  =  (37)(37)1( 3 \cdot 7 )\cdot( 3 \cdot 7 )\cdot1
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 33
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 10.369760452363677327351334332217861112 10.369760452363677327351334332217861112
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

10.369760452L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/211.1669700.479802441322.64575110.369760452\begin{aligned}10.369760452 \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 1.166970 \cdot 0.479802 \cdot 441 } { {3^2 \cdot 2.645751} } \\ & \approx 10.369760452 \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))
(a)(a) 22 2121 I21I_{21} Split multiplicative 1-1 11 2121 2121
(a+1)(-a+1) 22 2121 I21I_{21} Split multiplicative 1-1 11 2121 2121
(3)(3) 99 11 IIII Additive 11 44 44 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
33 3B.1.1
77 7B.2.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 3, 7 and 21.
Its isogeny class 26244.2-l consists of curves linked by isogenies of degrees dividing 21.

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 162.c2
Q\Q 7938.x2