Properties

Label 4.4.2304.1-9.1-a2
Base field Q(2,3)\Q(\sqrt{2}, \sqrt{3})
Conductor norm 9 9
CM no
Base change no
Q-curve yes
Torsion order 2 2
Rank 0 0

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field Q(2,3)\Q(\sqrt{2}, \sqrt{3})

Generator aa, with minimal polynomial x44x2+1 x^{4} - 4 x^{2} + 1 ; class number 11.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -4, 0, 1]))
 
gp: K = nfinit(Polrev([1, 0, -4, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, 0, 1]);
 

Weierstrass equation

y2+(a3+a24a2)xy+(a3+a24a1)y=x3+(a3+a23a3)x2+(2a3+a2+4a14)x6a3+2a2+24a7{y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(-2a^{3}+a^{2}+4a-14\right){x}-6a^{3}+2a^{2}+24a-7
sage: E = EllipticCurve([K([-2,-4,1,1]),K([-3,-3,1,1]),K([-1,-4,1,1]),K([-14,4,1,-2]),K([-7,24,2,-6])])
 
gp: E = ellinit([Polrev([-2,-4,1,1]),Polrev([-3,-3,1,1]),Polrev([-1,-4,1,1]),Polrev([-14,4,1,-2]),Polrev([-7,24,2,-6])], K);
 
magma: E := EllipticCurve([K![-2,-4,1,1],K![-3,-3,1,1],K![-1,-4,1,1],K![-14,4,1,-2],K![-7,24,2,-6]]);
 

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
(a3+14a2+92a+14:158a3118a2+538a+258:1)\left(-a^{3} + \frac{1}{4} a^{2} + \frac{9}{2} a + \frac{1}{4} : -\frac{15}{8} a^{3} - \frac{11}{8} a^{2} + \frac{53}{8} a + \frac{25}{8} : 1\right)0022

Invariants

Conductor: N\frak{N} = (a22)(a^2-2) = (a22)(a^2-2)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 9 9 = 99
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = a2+2-a^2+2
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (a2+2)(-a^2+2) = (a22)(a^2-2)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 9 9 = 99
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 4679515283a323397576403a+382083912 \frac{467951528}{3} a^{3} - \frac{2339757640}{3} a + 382083912
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) 285.83292677657224671119127374294579967 285.83292677657224671119127374294579967
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 1 1
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 22
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 1.48871316029465 1.48871316029465
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

1.488713160L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/21285.832927112248.0000001.488713160\displaystyle 1.488713160 \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 285.832927 \cdot 1 \cdot 1 } { {2^2 \cdot 48.000000} } \approx 1.488713160

Local data at primes of bad reduction

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

This elliptic curve is semistable. There is only one prime 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))
(a22)(a^2-2) 99 11 I1I_{1} Non-split multiplicative 11 11 11 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
55 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 4, 5, 10 and 20.
Its isogeny class 9.1-a consists of curves linked by isogenies of degrees dividing 20.

Base change

This elliptic curve is a Q\Q-curve.

It is not the base change of an elliptic curve defined over any subfield.