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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

\({y}^2+{x}{y}={x}^{3}+\left(-i+1\right){x}^{2}+\left(-15i+13\right){x}+3i+25\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([1,0]),K([1,-1]),K([0,0]),K([13,-15]),K([25,3])])
 
Copy content gp:E = ellinit([Polrev([1,0]),Polrev([1,-1]),Polrev([0,0]),Polrev([13,-15]),Polrev([25,3])], K);
 
Copy content magma:E := EllipticCurve([K![1,0],K![1,-1],K![0,0],K![13,-15],K![25,3]]);
 
Copy content oscar:E = elliptic_curve([K([1,0]),K([1,-1]),K([0,0]),K([13,-15]),K([25,3])])
 

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

trivial

Invariants

Conductor: $\frak{N}$ = \((25i-125)\) = \((i+1)\cdot(-i-2)^{2}\cdot(2i+1)^{2}\cdot(2i+3)\)
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: $N(\frak{N})$ = \( 16250 \) = \(2\cdot5^{2}\cdot5^{2}\cdot13\)
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: $\Delta$ = $-16625i+177375$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-16625i+177375)\) = \((i+1)\cdot(-i-2)^{10}\cdot(2i+1)^{3}\cdot(2i+3)\)
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: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 31738281250 \) = \(2\cdot5^{10}\cdot5^{3}\cdot13\)
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: $j$ = \( \frac{74877}{26} i + \frac{83939}{26} \)
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: $r_{\mathrm{an}}$= \( 0 \)
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: $r$ = \(0\)
Regulator: $\mathrm{Reg}(E/K)$ = \( 1 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ = \( 1 \)
Global period: $\Omega(E/K)$≈ \( 3.4926730845387729648598342725541308502 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 2 \)  =  \(1\cdot1\cdot2\cdot1\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(1\)
Special value: $L^{(r)}(E/K,1)/r!$ ≈\( 3.4926730845387729648598342725541308502 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}3.492673085 \approx L(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.492673 \cdot 1 \cdot 2 } { {1^2 \cdot 2.000000} } \\ & \approx 3.492673085 \end{aligned}$$

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

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((i+1)\) \(2\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-i-2)\) \(5\) \(1\) \(II^{*}\) Additive \(1\) \(2\) \(10\) \(0\)
\((2i+1)\) \(5\) \(2\) \(III\) Additive \(-1\) \(2\) \(3\) \(0\)
\((2i+3)\) \(13\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 16250.6-m consists of curves linked by isogenies of degree 5.

Base change

This elliptic curve is not a \(\Q\)-curve.

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