Properties

Label 2.2.12.1-3025.1-d4
Base field Q(3)\Q(\sqrt{3})
Conductor norm 3025 3025
CM no
Base change yes
Q-curve yes
Torsion order 2 2
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+axy=x359x190{y}^2+a{x}{y}={x}^{3}-59{x}-190
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,0]),K([-59,0]),K([-190,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-59,0]),Polrev([-190,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,0],K![-59,0],K![-190,0]]);
 

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
(5a+4:12a+25:1)\left(-5 a + 4 : -12 a + 25 : 1\right)0.329657945005665710769639655954648665460.32965794500566571076963965595464866546\infty
(194:198a:1)\left(-\frac{19}{4} : \frac{19}{8} a : 1\right)0022

Invariants

Conductor: N\frak{N} = (55)(55) = (2a+1)(2a+1)(5)(-2a+1)\cdot(2a+1)\cdot(5)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 3025 3025 = 11112511\cdot11\cdot25
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 68756875
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (6875)(6875) = (2a+1)(2a+1)(5)4(-2a+1)\cdot(2a+1)\cdot(5)^{4}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 47265625 47265625 = 111125411\cdot11\cdot25^{4}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 229305093216875 \frac{22930509321}{6875}
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) 0.32965794500566571076963965595464866546 0.32965794500566571076963965595464866546
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 0.659315890011331421539279311909297330920 0.659315890011331421539279311909297330920
Global period: Ω(E/K)\Omega(E/K) 2.9627822853300284644044638908933039909 2.9627822853300284644044638908933039909
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 4 4  =  11221\cdot1\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! 2.2556029313066144373456657553803586646 2.2556029313066144373456657553803586646
Analytic order of Ш: Шan{}_{\mathrm{an}}= 4 4 (rounded)

BSD formula

2.255602931L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/242.9627820.6593164223.4641022.255602931\displaystyle 2.255602931 \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 2.962782 \cdot 0.659316 \cdot 4 } { {2^2 \cdot 3.464102} } \approx 2.255602931

Local data at primes of bad reduction

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

This elliptic curve is 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))
(2a+1)(-2a+1) 1111 11 I1I_{1} Split multiplicative 1-1 11 11 11
(2a+1)(2a+1) 1111 11 I1I_{1} Split multiplicative 1-1 11 11 11
(5)(5) 2525 44 I4I_{4} Split multiplicative 1-1 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 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2 and 4.
Its isogeny class 3025.1-d consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a Q\Q-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
Q\Q 495.a1
Q\Q 880.h1