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Properties

Label 2.2.28.1-256.1-j8
Base field Q(7)\Q(\sqrt{7})
Conductor norm 256 256
CM yes (−112-112)
Base change no
Q-curve yes
Torsion order 4 4
Rank 0 0

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

Generator aa, with minimal polynomial x2−7 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=x3+(−240a−725)x+3698a+9520{y}^2={x}^{3}+\left(-240a-725\right){x}+3698a+9520
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-725,-240]),K([9520,3698])])
 
Copy content gp:E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-725,-240]),Polrev([9520,3698])], K);
 
Copy content magma:E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-725,-240],K![9520,3698]]);
 

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

Z/4Z\Z/{4}\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(−a+24:24a−64:1)\left(-a + 24 : 24 a - 64 : 1\right)0044

Invariants

Conductor: N\frak{N} = (16)(16) = (a+3)8(a+3)^{8}
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}) = 256 256 = 282^{8}
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 = −4096-4096
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (−4096)(-4096) = (a+3)24(a+3)^{24}
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) = 16777216 16777216 = 2242^{24}
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 = 51954490735875a+137458661985000 51954490735875 a + 137458661985000
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[−28]\Z[\sqrt{-28}]    (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) = N(U(1))N(\mathrm{U}(1))

BSD invariants

Analytic rank: ranr_{\mathrm{an}}= 0 0
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content 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)≈ 6.5409647643812237351895407197832776641 6.5409647643812237351895407197832776641
Tamagawa product: ∏pcp\prod_{\frak{p}}c_{\frak{p}}= 4 4
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 44
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! ≈1.2361261500706366841245492420460944111 1.2361261500706366841245492420460944111
Analytic order of Ш: Шan{}_{\mathrm{an}}= 4 4 (rounded)

BSD formula

1.236126150≈L(E/K,1)=?#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp#E(K)tor2⋅∣dK∣1/2≈4⋅6.540965⋅1⋅442⋅5.291503≈1.236126150\begin{aligned}1.236126150 \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 6.540965 \cdot 1 \cdot 4 } { {4^2 \cdot 5.291503} } \\ & \approx 1.236126150 \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 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))
(a+3)(a+3) 22 44 I12∗I_{12}^{*} Additive 11 88 2424 00

Galois Representations

The mod p p Galois Representation has maximal image for all primes p<1000 p < 1000 .

The image is a Borel subgroup if p∈{2,7}p\in \{ 2, 7\}, the normalizer of a split Cartan subgroup if (−7p)=+1\left(\frac{ -7 }{p}\right)=+1 or the normalizer of a nonsplit Cartan subgroup if (−7p)=−1\left(\frac{ -7 }{p}\right)=-1.

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2, 4, 7, 14 and 28.
Its isogeny class 256.1-j consists of curves linked by isogenies of degrees dividing 28.

Base change

This elliptic curve is a Q\Q-curve.

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