Properties

Label 2.2.8.1-3600.1-n3
Base field Q(2)\Q(\sqrt{2})
Conductor norm 3600 3600
CM no
Base change no
Q-curve no
Torsion order 4 4
Rank 0 0

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Base field Q(2)\Q(\sqrt{2})

Generator aa, with minimal polynomial x22 x^{2} - 2 ; class number 11.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 1]))
 
gp: K = nfinit(Polrev([-2, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
 

Weierstrass equation

y2+axy=x3+(a+1)x2+(25a35)x+80a116{y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(25a-35\right){x}+80a-116
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([0,0]),K([-35,25]),K([-116,80])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,1]),Polrev([0,0]),Polrev([-35,25]),Polrev([-116,80])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,1],K![0,0],K![-35,25],K![-116,80]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(a72:74a1:1)\left(a - \frac{7}{2} : \frac{7}{4} a - 1 : 1\right)0022
(4a+4:2a+4:1)\left(-4 a + 4 : -2 a + 4 : 1\right)0022

Invariants

Conductor: N\frak{N} = (60)(60) = (a)4(3)(5)(a)^{4}\cdot(3)\cdot(5)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 3600 3600 = 249252^{4}\cdot9\cdot25
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 900900
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (900)(900) = (a)4(3)2(5)2(a)^{4}\cdot(3)^{2}\cdot(5)^{2}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 810000 810000 = 24922522^{4}\cdot9^{2}\cdot25^{2}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 583258208225a+27576545675 -\frac{583258208}{225} a + \frac{275765456}{75}
sage: E.j_invariant()
 
gp: E.j
 
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)
sage: E.has_cm(), E.cm_discriminant()
 
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
sage: E.rank()
 
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) 6.3111528611510929367633129349222374096 6.3111528611510929367633129349222374096
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 4 4  =  1221\cdot2\cdot2
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 44
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 2.2313294926124096117547924941649565651 2.2313294926124096117547924941649565651
Analytic order of Ш: Шan{}_{\mathrm{an}}= 4 4 (rounded)

BSD formula

2.231329493L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/246.31115314422.8284272.231329493\displaystyle 2.231329493 \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{ 4 \cdot 6.311153 \cdot 1 \cdot 4 } { {4^2 \cdot 2.828427} } \approx 2.231329493

Local data at primes of bad reduction

sage: E.local_data()
 
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 11 IIII Additive 11 44 44 00
(3)(3) 99 22 I2I_{2} Split multiplicative 1-1 11 22 22
(5)(5) 2525 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 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2.
Its isogeny class 3600.1-n consists of curves linked by isogenies of degrees dividing 4.

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.