Properties

Label 2.2.28.1-567.1-m2
Base field Q(7)\Q(\sqrt{7})
Conductor norm 567 567
CM no
Base change no
Q-curve no
Torsion order 2 2
Rank 1 1

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

Generator aa, with minimal polynomial x27 x^{2} - 7 ; class number 11.

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, 0, 1]))
 
Copy content gp:K = nfinit(Polrev([-7, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);
 

Weierstrass equation

y2+(a+1)xy+ay=x3+(a+1)x2+(148a380)x+1439a3796{y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(148a-380\right){x}+1439a-3796
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,1]),K([1,1]),K([0,1]),K([-380,148]),K([-3796,1439])])
 
Copy content gp:E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,1]),Polrev([-380,148]),Polrev([-3796,1439])], K);
 
Copy content magma:E := EllipticCurve([K![1,1],K![1,1],K![0,1],K![-380,148],K![-3796,1439]]);
 

This is a global minimal model.

Copy content comment:Test whether it is a global minimal model
 
Copy content 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
(3a10:a:1)\left(3 a - 10 : a : 1\right)0.878819682589044832761034824544528759400.87881968258904483276103482454452875940\infty
(132a+15:194a+614:1)\left(-\frac{13}{2} a + 15 : -\frac{19}{4} a + \frac{61}{4} : 1\right)0022

Invariants

Conductor: N\frak{N} = (9a)(9a) = (a+2)2(a2)2(a)(-a+2)^{2}\cdot(-a-2)^{2}\cdot(a)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Conductor norm: N(N)N(\frak{N}) = 567 567 = 323273^{2}\cdot3^{2}\cdot7
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Discriminant: Δ\Delta = 18522a9261-18522a-9261
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (18522a9261)(-18522a-9261) = (a+2)3(a2)6(a)6(-a+2)^{3}\cdot(-a-2)^{6}\cdot(a)^{6}
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Discriminant norm: N(Dmin)=N(Δ)N(\frak{D}_{\mathrm{min}}) = N(\Delta) = 2315685267 -2315685267 = 333676-3^{3}\cdot3^{6}\cdot7^{6}
Copy content comment:Compute the norm of the discriminant
 
Copy content sage:E.discriminant().norm()
 
Copy content gp:norm(E.disc)
 
Copy content magma:Norm(Discriminant(E));
 
j-invariant: jj = 1204736343a+3558976343 -\frac{1204736}{343} a + \frac{3558976}{343}
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content 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)
Copy content comment:Test for Complex Multiplication
 
Copy content sage:E.has_cm(), E.cm_discriminant()
 
Copy content 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
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: rr = 11
Regulator: Reg(E/K)\mathrm{Reg}(E/K) 0.87881968258904483276103482454452875940 0.87881968258904483276103482454452875940
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 1.75763936517808966552206964908905751880 1.75763936517808966552206964908905751880
Global period: Ω(E/K)\Omega(E/K) 4.4101128779332199173848050291868327107 4.4101128779332199173848050291868327107
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 8 8  =  2222\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! 2.9297492801828298741848710359144752013 2.9297492801828298741848710359144752013
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

2.929749280L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/214.4101131.7576398225.2915032.929749280\begin{aligned}2.929749280 \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 4.410113 \cdot 1.757639 \cdot 8 } { {2^2 \cdot 5.291503} } \\ & \approx 2.929749280 \end{aligned}

Local data at primes of bad reduction

Copy content comment:Compute the local reduction data at primes of bad reduction
 
Copy content sage:E.local_data()
 
Copy content 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+2)(-a+2) 33 22 IIIIII Additive 11 22 33 00
(a2)(-a-2) 33 22 I0I_0^{*} Additive 1-1 22 66 00
(a)(a) 77 22 I6I_{6} Non-split multiplicative 11 11 66 66

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
33 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 3 and 6.
Its isogeny class 567.1-m consists of curves linked by isogenies of degrees dividing 6.

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.