Properties

Label 2.0.7.1-9072.1-a3
Base field Q(7)\Q(\sqrt{-7})
Conductor norm 9072 9072
CM no
Base change no
Q-curve no
Torsion order 4 4
Rank 1 1

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Base field Q(7)\Q(\sqrt{-7})

Generator aa, with minimal polynomial x2x+2 x^{2} - x + 2 ; class number 11.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
 
gp: K = nfinit(Polrev([2, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

y2+axy=x3+(a1)x2+(3a9)x+35a14{y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a-9\right){x}+35a-14
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,0]),K([-9,-3]),K([-14,35])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([-1,-1]),Polrev([0,0]),Polrev([-9,-3]),Polrev([-14,35])], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,0],K![-9,-3],K![-14,35]]);
 

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
(14a94:178a298:1)\left(-\frac{1}{4} a - \frac{9}{4} : -\frac{17}{8} a - \frac{29}{8} : 1\right)0.902139753122063296073009131376127872950.90213975312206329607300913137612787295\infty
(a3:2a1:1)\left(-a - 3 : 2 a - 1 : 1\right)0022
(2a:a+2:1)\left(2 a : -a + 2 : 1\right)0022

Invariants

Conductor: N\frak{N} = (9a+90)(9a+90) = (a)4(2a+1)(3)2(a)^{4}\cdot(-2a+1)\cdot(3)^{2}
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 9072 9072 = 247922^{4}\cdot7\cdot9^{2}
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: Δ\Delta = 137781a+642978137781a+642978
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (137781a+642978)(137781a+642978) = (a)8(2a+1)2(3)8(a)^{8}\cdot(-2a+1)^{2}\cdot(3)^{8}
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 539978068224 539978068224 = 2872982^{8}\cdot7^{2}\cdot9^{8}
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: jj = 400921a+331063 \frac{4009}{21} a + \frac{3310}{63}
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) 0.90213975312206329607300913137612787295 0.90213975312206329607300913137612787295
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 1.80427950624412659214601826275225574590 1.80427950624412659214601826275225574590
Global period: Ω(E/K)\Omega(E/K) 2.9271621459193486657486515934021316474 2.9271621459193486657486515934021316474
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 16 16  =  22222\cdot2\cdot2^{2}
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 44
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 1.9961886248525515495016296397597168588 1.9961886248525515495016296397597168588
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

1.996188625L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/212.9271621.80428016422.6457511.996188625\displaystyle 1.996188625 \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 2.927162 \cdot 1.804280 \cdot 16 } { {4^2 \cdot 2.645751} } \approx 1.996188625

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)(a) 22 22 I0I_0^{*} Additive 11 44 88 00
(2a+1)(-2a+1) 77 22 I2I_{2} Non-split multiplicative 11 11 22 22
(3)(3) 99 44 I2I_{2}^{*} Additive 11 22 88 22

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.
Its isogeny class 9072.1-a consists of curves linked by isogenies of degrees dividing 4.

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.