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Properties

Label 4.4.12400.1-5.2-a1
Base field 4.4.12400.1
Conductor norm \( 5 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.12400.1

Generator \(a\), with minimal polynomial \( x^{4} - 12 x^{2} + 31 \); class number \(1\).

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([31, 0, -12, 0, 1]))
 
Copy content gp:K = nfinit(Polrev([31, 0, -12, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 0, -12, 0, 1]);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx([31, 0, -12, 0, 1]))
 

Weierstrass equation

\({y}^2+\left(\frac{1}{2}a^{3}-\frac{7}{2}a\right){x}{y}+\left(\frac{1}{2}a^{2}+a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a+1\right){x}^{2}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a+8\right){x}-\frac{3}{2}a^{3}-2a^{2}+\frac{13}{2}a+9\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,-7/2,0,1/2]),K([1,5/2,0,-1/2]),K([-5/2,1,1/2,0]),K([8,5/2,0,-1/2]),K([9,13/2,-2,-3/2])])
 
Copy content gp:E = ellinit([Polrev([0,-7/2,0,1/2]),Polrev([1,5/2,0,-1/2]),Polrev([-5/2,1,1/2,0]),Polrev([8,5/2,0,-1/2]),Polrev([9,13/2,-2,-3/2])], K);
 
Copy content magma:E := EllipticCurve([K![0,-7/2,0,1/2],K![1,5/2,0,-1/2],K![-5/2,1,1/2,0],K![8,5/2,0,-1/2],K![9,13/2,-2,-3/2]]);
 
Copy content oscar:E = elliptic_curve([K([0,-7/2,0,1/2]),K([1,5/2,0,-1/2]),K([-5/2,1,1/2,0]),K([8,5/2,0,-1/2]),K([9,13/2,-2,-3/2])])
 

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: $\frak{N}$ = \((-1/2a^3-3/2a^2+3/2a+11/2)\) = \((-1/2a^3-3/2a^2+3/2a+11/2)\)
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: $N(\frak{N})$ = \( 5 \) = \(5\)
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: $\Delta$ = $-1/2a^3-5a^2-27/2a+24$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-1/2a^3-5a^2-27/2a+24)\) = \((-1/2a^3-3/2a^2+3/2a+11/2)^{9}\)
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: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( -1953125 \) = \(-5^{9}\)
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: $j$ = \( -\frac{102109}{6250} a^{3} - \frac{1063917}{3125} a^{2} + \frac{972919}{6250} a + \frac{8947842}{3125} \)
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: $\Omega(E/K)$≈ \( 217.01636668283842223207841537059219749 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 1 \)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(1\)
Special value: $L^{(r)}(E/K,1)/r!$ ≈\( 1.94886450414123 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}1.948864504 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 217.016367 \cdot 1 \cdot 1 } { {1^2 \cdot 111.355287} } \\ & \approx 1.948864504 \end{aligned}$$

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 is only one prime $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-1/2a^3-3/2a^2+3/2a+11/2)\) \(5\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 5.2-a consists of curves linked by isogenies of degree 3.

Base change

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

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