Properties

Label 3.1.23.1-136.1-A2
Base field 3.1.23.1
Conductor norm \( 136 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

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Base field 3.1.23.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -1, 1]))
 
gp: K = nfinit(Polrev([1, 0, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -1, 1]);
 

Weierstrass equation

\({y}^2+a^{2}{x}{y}+\left(a^{2}+1\right){y}={x}^{3}-{x}^{2}+\left(-a^{2}+26a+21\right){x}-88a^{2}+13a+59\)
sage: E = EllipticCurve([K([0,0,1]),K([-1,0,0]),K([1,0,1]),K([21,26,-1]),K([59,13,-88])])
 
gp: E = ellinit([Polrev([0,0,1]),Polrev([-1,0,0]),Polrev([1,0,1]),Polrev([21,26,-1]),Polrev([59,13,-88])], K);
 
magma: E := EllipticCurve([K![0,0,1],K![-1,0,0],K![1,0,1],K![21,26,-1],K![59,13,-88]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(3 a^{2} - 1 : -a^{2} + a + 1 : 1\right)$$0$$3$

Invariants

Conductor: $\frak{N}$ = \((-2a^2-4)\) = \((2)\cdot(-a^2-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 136 \) = \(8\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $-40a^2-80a-96$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-40a^2-80a-96)\) = \((2)^{3}\cdot(-a^2-2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( -2515456 \) = \(-8^{3}\cdot17^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( \frac{214693602205}{19652} a^{2} + \frac{1064389027655}{39304} a + \frac{69858641986}{4913} \)
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: $r_{\mathrm{an}}$= \( 0 \)
sage: E.rank()
 
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: $\Omega(E/K)$ \( 17.595531818548944326107374447439471226 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 1 \)  =  \(1\cdot1\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(3\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 0.40765800079637202880182367092307604747 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 0.407658001 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 17.595532 \cdot 1 \cdot 1 } { {3^2 \cdot 4.795832} } \approx 0.407658001$

Local data at primes of bad reduction

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

This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((2)\) \(8\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((-a^2-2)\) \(17\) \(1\) \(I_{3}\) Non-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 136.1-A consists of curves linked by isogenies of degrees dividing 9.

Base change

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

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