Properties

Label 2.2.56.1-10.1-c2
Base field Q(14)\Q(\sqrt{14})
Conductor norm 10 10
CM no
Base change no
Q-curve no
Torsion order 1 1
Rank 1 1

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Base field Q(14)\Q(\sqrt{14})

Generator aa, with minimal polynomial x214 x^{2} - 14 ; class number 11.

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))
 
Copy content gp:K = nfinit(Polrev([-14, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
 

Weierstrass equation

y2+(a+1)xy=x3+(1285a4941)x48650a+182431{y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(1285a-4941\right){x}-48650a+182431
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([-4941,1285]),K([182431,-48650])])
 
Copy content gp:E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-4941,1285]),Polrev([182431,-48650])], K);
 
Copy content magma:E := EllipticCurve([K![1,1],K![0,0],K![0,0],K![-4941,1285],K![182431,-48650]]);
 

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\Z

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(4a+20:43a+149:1)\left(-4 a + 20 : -43 a + 149 : 1\right)0.272002473373904783713542173453317126660.27200247337390478371354217345331712666\infty

Invariants

Conductor: N\frak{N} = (a+2)(a+2) = (a+4)(a+3)(-a+4)\cdot(-a+3)
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}) = 10 10 = 252\cdot5
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 = 2a+6-2a+6
Discriminant ideal: Dmin=(Δ)\frak{D}_{\mathrm{min}} = (\Delta) = (2a+6)(-2a+6) = (a+4)2(a+3)(-a+4)^{2}\cdot(-a+3)
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) = 20 -20 = 225-2^{2}\cdot5
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 = 711467655441806250310a+2662068208119956998910 \frac{7114676554418062503}{10} a + \frac{26620682081199569989}{10}
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) 0.27200247337390478371354217345331712666 0.27200247337390478371354217345331712666
Néron-Tate Regulator: RegNT(E/K)\mathrm{Reg}_{\mathrm{NT}}(E/K) 0.544004946747809567427084346906634253320 0.544004946747809567427084346906634253320
Global period: Ω(E/K)\Omega(E/K) 14.163134738548721329178436662472233718 14.163134738548721329178436662472233718
Tamagawa product: pcp\prod_{\frak{p}}c_{\frak{p}}= 2 2  =  212\cdot1
Torsion order: #E(K)tor\#E(K)_{\mathrm{tor}}= 11
Special value: L(r)(E/K,1)/r!L^{(r)}(E/K,1)/r! 2.0591985216127295013337256651752767828 2.0591985216127295013337256651752767828
Analytic order of Ш: Шan{}_{\mathrm{an}}= 1 1 (rounded)

BSD formula

2.059198522L(E/K,1)=?#Ш(E/K)Ω(E/K)RegNT(E/K)pcp#E(K)tor2dK1/2114.1631350.5440052127.4833152.059198522\begin{aligned}2.059198522 \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 14.163135 \cdot 0.544005 \cdot 2 } { {1^2 \cdot 7.483315} } \\ & \approx 2.059198522 \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 semistable. There are 2 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+4)(-a+4) 22 22 I2I_{2} Split multiplicative 1-1 11 22 22
(a+3)(-a+3) 55 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
55 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree dd for d=d= 5.
Its isogeny class 10.1-c consists of curves linked by isogenies of degree 5.

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.