Generator ϕ \phi ϕ , with minimal polynomial
x 2 − x − 1 x^{2} - x - 1 x 2 − x − 1 ; class number 1 1 1 .
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -1, 1]))
gp: K = nfinit(Polrev([-1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 1]);
y 2 + y = x 3 − x 2 + ( 14 ϕ + 5 ) x − 21 ϕ − 15 {y}^2+{y}={x}^{3}-{x}^{2}+\left(14\phi+5\right){x}-21\phi-15 y 2 + y = x 3 − x 2 + ( 1 4 ϕ + 5 ) x − 2 1 ϕ − 1 5
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([5,14]),K([-15,-21])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([5,14]),Polrev([-15,-21])], K);
magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![5,14],K![-15,-21]]);
This is a global minimal model .
sage: E.is_global_minimal_model()
trivial
Conductor :
N \frak{N} N
=
( 49 ) (49) ( 4 9 )
=
( 7 ) 2 (7)^{2} ( 7 ) 2
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
Conductor norm :
N ( N ) N(\frak{N}) N ( N )
=
2401 2401 2 4 0 1
=
4 9 2 49^{2} 4 9 2
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
Discriminant :
Δ \Delta Δ
=
− 343 -343 − 3 4 3
Discriminant ideal :
D m i n = ( Δ ) \frak{D}_{\mathrm{min}} = (\Delta) D m i n = ( Δ )
=
( − 343 ) (-343) ( − 3 4 3 )
=
( 7 ) 3 (7)^{3} ( 7 ) 3
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
Discriminant norm :
N ( D m i n ) = N ( Δ ) N(\frak{D}_{\mathrm{min}}) = N(\Delta) N ( D m i n ) = N ( Δ )
=
117649 117649 1 1 7 6 4 9
=
4 9 3 49^{3} 4 9 3
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
j-invariant :
j j j
=
52756480 ϕ − 85360640 52756480 \phi - 85360640 5 2 7 5 6 4 8 0 ϕ − 8 5 3 6 0 6 4 0
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
Endomorphism ring :
E n d ( E ) \mathrm{End}(E) E n d ( E )
=
Z \Z Z
Geometric endomorphism ring :
E n d ( E Q ‾ ) \mathrm{End}(E_{\overline{\Q}}) E n d ( E Q )
=
Z [ ( 1 + − 35 ) / 2 ] \Z[(1+\sqrt{-35})/2] Z [ ( 1 + − 3 5 ) / 2 ]
(potential complex multiplication )
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
Sato-Tate group :
S T ( E ) \mathrm{ST}(E) S T ( E )
=
N ( U ( 1 ) ) N(\mathrm{U}(1)) N ( U ( 1 ) )
Analytic rank :
r a n r_{\mathrm{an}} r a n =
0 0 0
sage: E.rank()
magma: Rank(E);
Mordell-Weil rank :
r r r
=
0 0 0
Regulator :
R e g ( E / K ) \mathrm{Reg}(E/K) R e g ( E / K )
=
1 1 1
Néron-Tate Regulator :
R e g N T ( E / K ) \mathrm{Reg}_{\mathrm{NT}}(E/K) R e g N T ( E / K )
=
1 1 1
Global period :
Ω ( E / K ) \Omega(E/K) Ω ( E / K ) ≈
2.4377555092472808403399055475038616336 2.4377555092472808403399055475038616336 2 . 4 3 7 7 5 5 5 0 9 2 4 7 2 8 0 8 4 0 3 3 9 9 0 5 5 4 7 5 0 3 8 6 1 6 3 3 6
Tamagawa product :
∏ p c p \prod_{\frak{p}}c_{\frak{p}} ∏ p c p =
2 2 2
Torsion order :
# E ( K ) t o r \#E(K)_{\mathrm{tor}} # E ( K ) t o r =
1 1 1
Special value :
L ( r ) ( E / K , 1 ) / r ! L^{(r)}(E/K,1)/r! L ( r ) ( E / K , 1 ) / r !
≈ 2.1803948124806148589184806540746934821 2.1803948124806148589184806540746934821 2 . 1 8 0 3 9 4 8 1 2 4 8 0 6 1 4 8 5 8 9 1 8 4 8 0 6 5 4 0 7 4 6 9 3 4 8 2 1
Analytic order of Ш :
Шa n {}_{\mathrm{an}} a n =
1 1 1 (rounded)
2.180394812 ≈ L ( E / K , 1 ) = ? # Ш ( E / K ) ⋅ Ω ( E / K ) ⋅ R e g N T ( E / K ) ⋅ ∏ p c p # E ( K ) t o r 2 ⋅ ∣ d K ∣ 1 / 2 ≈ 1 ⋅ 2.437756 ⋅ 1 ⋅ 2 1 2 ⋅ 2.236068 ≈ 2.180394812 \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} 2 . 1 8 0 3 9 4 8 1 2 ≈ L ( E / K , 1 ) = ? # E ( K ) t o r 2 ⋅ ∣ d K ∣ 1 / 2 # Ш ( E / K ) ⋅ Ω ( E / K ) ⋅ R e g N T ( E / K ) ⋅ ∏ p c p ≈ 1 2 ⋅ 2 . 2 3 6 0 6 8 1 ⋅ 2 . 4 3 7 7 5 6 ⋅ 1 ⋅ 2 ≈ 2 . 1 8 0 3 9 4 8 1 2
sage: E.local_data()
magma: LocalInformation(E);
This elliptic curve is not semistable .
There
is only one prime p \frak{p} p
of bad reduction .
This curve has non-trivial cyclic isogenies of degree d d d for d = d= d =
5, 7 and 35.
Its isogeny class
2401.1-b
consists of curves linked by isogenies of
degrees dividing 35.
This elliptic curve is a Q \Q Q -curve .
It is not the base change of an elliptic curve defined over any subfield.