Properties

Label 3.1.23.1-259.1-A2
Base field 3.1.23.1
Conductor norm \( 259 \)
CM no
Base change no
Q-curve no
Torsion order \( 9 \)
Rank \( 0 \)

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Base field 3.1.23.1

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

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

Weierstrass equation

\({y}^2+{y}={x}^{3}+a^{2}{x}^{2}+9a{x}+8a^{2}-15\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,0,0]),K([0,0,1]),K([1,0,0]),K([0,9,0]),K([-15,0,8])])
 
Copy content gp:E = ellinit([Polrev([0,0,0]),Polrev([0,0,1]),Polrev([1,0,0]),Polrev([0,9,0]),Polrev([-15,0,8])], K);
 
Copy content magma:E := EllipticCurve([K![0,0,0],K![0,0,1],K![1,0,0],K![0,9,0],K![-15,0,8]]);
 

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

\(\Z/{9}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(10 a^{2} - 21 a + 17 : 74 a^{2} - 131 a + 99 : 1\right)$$0$$9$

Invariants

Conductor: $\frak{N}$ = \((-4a^2+7a+1)\) = \((2a^2-a)\cdot(3a^2-a+1)\)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 259 \) = \(7\cdot37\)
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));
 
Discriminant: $\Delta$ = $-11800a^2+4160a-3211$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-11800a^2+4160a-3211)\) = \((2a^2-a)^{9}\cdot(3a^2-a+1)^{3}\)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( -2044031255371 \) = \(-7^{9}\cdot37^{3}\)
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));
 
j-invariant: $j$ = \( \frac{533423413596160}{2044031255371} a^{2} + \frac{14942973020037120}{2044031255371} a + \frac{5291457412308992}{2044031255371} \)
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(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: $\Omega(E/K)$ \( 7.4798711597035459106432971586883680384 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 27 \)  =  \(3^{2}\cdot3\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(9\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 0.51988698402933238686103269625191968865 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}0.519886984 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 7.479871 \cdot 1 \cdot 27 } { {9^2 \cdot 4.795832} } \\ & \approx 0.519886984 \end{aligned}$$

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

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((2a^2-a)\) \(7\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\((3a^2-a+1)\) \(37\) \(3\) \(I_{3}\) 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 and 9.
Its isogeny class 259.1-A consists of curves linked by isogenies of degrees dividing 27.

Base change

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

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