Properties

Label 2.2.12.1-2304.1-u1
Base field Q(3)\Q(\sqrt{3})
Conductor norm 2304 2304
CM no
Base change no
Q-curve no
Torsion order 2 2
Rank 0 0

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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=x3ax2+(56601488a98036652)x+303715215440a526050184174{y}^2={x}^{3}-a{x}^{2}+\left(56601488a-98036652\right){x}+303715215440a-526050184174
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([-98036652,56601488]),K([-526050184174,303715215440])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([-98036652,56601488]),Polrev([-526050184174,303715215440])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![-98036652,56601488],K![-526050184174,303715215440]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

Z/2Z\Z/{2}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(2461a4262:0:1)\left(2461 a - 4262 : 0 : 1\right)0022

Invariants

Conductor: N\frak{N} = (48)(48) = (a+1)8(a)2(a+1)^{8}\cdot(a)^{2}
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 2304 2304 = 28322^{8}\cdot3^{2}
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 1728a1728a
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (1728a)(1728a) = (a+1)12(a)7(a+1)^{12}\cdot(a)^{7}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 8957952 -8957952 = 21237-2^{12}\cdot3^{7}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 1660163a+95936 -\frac{166016}{3} a + 95936
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) 7.7830915032296410870543107897663836460 7.7830915032296410870543107897663836460
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 4 4  =  1221\cdot2^{2}
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 22
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 2.2467849872585611634644197290321034328 2.2467849872585611634644197290321034328
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

2.246784987L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/217.78309214223.4641022.246784987\displaystyle 2.246784987 \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 7.783092 \cdot 1 \cdot 4 } { {2^2 \cdot 3.464102} } \approx 2.246784987

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 11 I0I_0^{*} Additive 1-1 88 1212 00
(a)(a) 33 44 I1I_{1}^{*} Additive 1-1 22 77 11

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 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 4 and 8.
Its isogeny class 2304.1-u 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.