Properties

Label 2.2.24.1-375.2-a1
Base field Q(6)\Q(\sqrt{6})
Conductor norm 375 375
CM no
Base change no
Q-curve no
Torsion order 2 2
Rank 1 1

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Base field Q(6)\Q(\sqrt{6})

Generator aa, with minimal polynomial x26 x^{2} - 6 ; class number 11.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))
 
gp: K = nfinit(Polrev([-6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
 

Weierstrass equation

y2+(a+1)xy+y=x3+ax2+(a4)x+a1{y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(a-4\right){x}+a-1
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([1,0]),K([-4,1]),K([-1,1])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([1,0]),Polrev([-4,1]),Polrev([-1,1])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![1,0],K![-4,1],K![-1,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(a+3:a2:1)\left(-a + 3 : a - 2 : 1\right)1.04062514468901886137206676043159655241.0406251446890188613720667604315965524\infty
(a3:2a+4:1)\left(-a - 3 : 2 a + 4 : 1\right)0022

Invariants

Conductor: N\frak{N} = (10a15)(-10a-15) = (a+3)(a1)(a+1)2(a+3)\cdot(-a-1)\cdot(-a+1)^{2}
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 375 375 = 35523\cdot5\cdot5^{2}
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 5350a5025-5350a-5025
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (5350a5025)(-5350a-5025) = (a+3)(a1)2(a+1)9(a+3)\cdot(-a-1)^{2}\cdot(-a+1)^{9}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 146484375 -146484375 = 35259-3\cdot5^{2}\cdot5^{9}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 20805875a+3355125 -\frac{208058}{75} a + \frac{33551}{25}
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) 1.0406251446890188613720667604315965524 1.0406251446890188613720667604315965524
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 2.0812502893780377227441335208631931048 2.0812502893780377227441335208631931048
Global period: Ω(E/K)\Omega(E/K) 10.532504238390417681239044847045936315 10.532504238390417681239044847045936315
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 4 4  =  1221\cdot2\cdot2
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 22
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 4.4745599687873097608045487742521184675 4.4745599687873097608045487742521184675
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

4.474559969L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/2110.5325042.0812504224.8989794.474559969\displaystyle 4.474559969 \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 10.532504 \cdot 2.081250 \cdot 4 } { {2^2 \cdot 4.898979} } \approx 4.474559969

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+3)(a+3) 33 11 I1I_{1} Split multiplicative 1-1 11 11 11
(a1)(-a-1) 55 22 I2I_{2} Split multiplicative 1-1 11 22 22
(a+1)(-a+1) 55 22 IIIIII^{*} Additive 1-1 22 99 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
22 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2.
Its isogeny class 375.2-a consists of curves linked by isogenies of degree 2.

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.