Properties

Label 2.2.12.1-768.1-p3
Base field Q(3)\Q(\sqrt{3})
Conductor norm 768 768
CM no
Base change no
Q-curve no
Torsion order 4 4
Rank 1 1

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Base field Q(3)\Q(\sqrt{3})

Generator aa, with minimal polynomial x23 x^{2} - 3 ; class number 11.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
 
gp: K = nfinit(Polrev([-3, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

y2=x3+(a1)x2+(2286a3957)x77553a+134325{y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(2286a-3957\right){x}-77553a+134325
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-3957,2286]),K([134325,-77553])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([0,0]),Polrev([-3957,2286]),Polrev([134325,-77553])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-3957,2286],K![134325,-77553]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(3a+6:123a213:1)\left(-3 a + 6 : 123 a - 213 : 1\right)0.423668005679459396658963573531204878110.42366800567945939665896357353120487811\infty
(14a+25:0:1)\left(-14 a + 25 : 0 : 1\right)0022
(30a51:0:1)\left(30 a - 51 : 0 : 1\right)0022

Invariants

Conductor: N\frak{N} = (16a)(16a) = (a+1)8(a)(a+1)^{8}\cdot(a)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 768 768 = 2832^{8}\cdot3
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 4608-4608
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (4608)(-4608) = (a+1)18(a)4(a+1)^{18}\cdot(a)^{4}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 21233664 21233664 = 218342^{18}\cdot3^{4}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 11220889a+19898089 -\frac{1122088}{9} a + \frac{1989808}{9}
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}}= 1 1
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: rr = 11
Regulator: Reg(E/K)\mathrm{Reg}(E/K) 0.42366800567945939665896357353120487811 0.42366800567945939665896357353120487811
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 0.847336011358918793317927147062409756220 0.847336011358918793317927147062409756220
Global period: Ω(E/K)\Omega(E/K) 17.976744299323390120705109272463816060 17.976744299323390120705109272463816060
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 8 8  =  2222^{2}\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! 2.1985993056964826472999359845881297854 2.1985993056964826472999359845881297854
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

2.198599306L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/2117.9767440.8473368423.4641022.198599306\displaystyle 2.198599306 \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 17.976744 \cdot 0.847336 \cdot 8 } { {4^2 \cdot 3.464102} } \approx 2.198599306

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 

This elliptic curve is not semistable. There are 2 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+1)(a+1) 22 44 I6I_{6}^{*} Additive 1-1 88 1818 00
(a)(a) 33 22 I4I_{4} Non-split multiplicative 11 11 44 44

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 and 4.
Its isogeny class 768.1-p 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.