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Properties

Label 2.0.4.1-8281.2-b3
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 8281 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 3 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{-1}) \)

Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).

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

Weierstrass equation

\({y}^2+{y}={x}^{3}+{x}^{2}+13{x}+42\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,0]),K([1,0]),K([1,0]),K([13,0]),K([42,0])])
 
Copy content gp:E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([1,0]),Polrev([13,0]),Polrev([42,0])], K);
 
Copy content magma:E := EllipticCurve([K![0,0],K![1,0],K![1,0],K![13,0],K![42,0]]);
 
Copy content oscar:E = elliptic_curve([K([0,0]),K([1,0]),K([1,0]),K([13,0]),K([42,0])])
 

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

\(\Z \oplus \Z \oplus \Z/{3}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-2 : 3 : 1\right)$$0.35308169546971684849771789725392435252$$\infty$
$\left(-7 i - 9 : -21 i + 31 : 1\right)$$0.44685529576190413736168851394534197034$$\infty$
$\left(0 : -7 : 1\right)$$0$$3$

Invariants

Conductor: $\frak{N}$ = \((91)\) = \((-3i-2)\cdot(2i+3)\cdot(7)\)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Conductor norm: $N(\frak{N})$ = \( 8281 \) = \(13\cdot13\cdot49\)
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(K, ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Copy content oscar:norm(conductor(E))
 
Discriminant: $\Delta$ = $-753571$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-753571)\) = \((-3i-2)^{3}\cdot(2i+3)^{3}\cdot(7)^{3}\)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 567869252041 \) = \(13^{3}\cdot13^{3}\cdot49^{3}\)
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));
 
Copy content oscar:norm(discriminant(E))
 
j-invariant: $j$ = \( \frac{224755712}{753571} \)
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z\)    (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: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 2 \)
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: $r$ = \(2\)
Regulator: $\mathrm{Reg}(E/K)$ ≈ \( 0.12660975453829242368300035779115648168 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ ≈ \( 0.506439018153169694732001431164625926720 \)
Global period: $\Omega(E/K)$≈ \( 2.9199070570469477852856741917215785260 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 27 \)  =  \(3\cdot3\cdot3\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(3\)
Special value: $L^{(r)}(E/K,1)/r!$ ≈\( 2.2181322946040512332765740514834917924 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}2.218132295 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2.919907 \cdot 0.506439 \cdot 27 } { {3^2 \cdot 2.000000} } \\ & \approx 2.218132295 \end{aligned}$$

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 3 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-3i-2)\) \(13\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((2i+3)\) \(13\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((7)\) \(49\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 8281.2-b consists of curves linked by isogenies of degrees dividing 9.

Base change

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

Base field Curve
\(\Q\) 91.b3
\(\Q\) 1456.k3