Properties

Label 2.2.13.1-81.4-c2
Base field \(\Q(\sqrt{13}) \)
Conductor norm \( 81 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{13}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, -1, 1]))
 
gp: K = nfinit(Polrev([-3, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -1, 1]);
 

Weierstrass equation

\({y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a-5\right){x}-a+1\)
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([0,1]),K([-5,2]),K([1,-1])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,1]),Polrev([0,1]),Polrev([-5,2]),Polrev([1,-1])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,1],K![0,1],K![-5,2],K![1,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-1 : -a + 2 : 1\right)$$0.024560134896961457895628856752853922426$$\infty$

Invariants

Conductor: $\frak{N}$ = \((5a-2)\) = \((-a+1)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 81 \) = \(3^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $-64a+301$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-64a+301)\) = \((-a+1)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 59049 \) = \(3^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( 20480 a + 28672 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z\)    (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: $r$ = \(1\)
Regulator: $\mathrm{Reg}(E/K)$ \( 0.024560134896961457895628856752853922426 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 0.04912026979392291579125771350570784485200 \)
Global period: $\Omega(E/K)$ \( 24.799239239362419698101330763125579823 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 3 \)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(1\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 1.0135581738174795948613656275892947792 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 1.013558174 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 24.799239 \cdot 0.049120 \cdot 3 } { {1^2 \cdot 3.605551} } \approx 1.013558174$

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 

This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-a+1)\) \(3\) \(3\) \(IV^{*}\) Additive \(-1\) \(4\) \(10\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cn
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 81.4-c 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.