Properties

Label 2.2.57.1-228.1-h6
Base field \(\Q(\sqrt{57}) \)
Conductor norm \( 228 \)
CM no
Base change no
Q-curve yes
Torsion order \( 6 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{57}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 14 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -1, 1]))
 
gp: K = nfinit(Polrev([-14, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2126a-6960\right){x}+99563a+326059\)
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([1,1]),K([-6960,-2126]),K([326059,99563])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,1]),Polrev([1,1]),Polrev([-6960,-2126]),Polrev([326059,99563])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,1],K![1,1],K![-6960,-2126],K![326059,99563]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z/{6}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(27 a + 89 : 317 a + 1039 : 1\right)$$0$$6$

Invariants

Conductor: $\frak{N}$ = \((-4a+2)\) = \((a-4)\cdot(a+3)\cdot(4a+13)\cdot(10a-43)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 228 \) = \(2\cdot2\cdot3\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $5880a+19632$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((5880a+19632)\) = \((a-4)^{12}\cdot(a+3)^{3}\cdot(4a+13)^{3}\cdot(10a-43)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 16809984 \) = \(2^{12}\cdot2^{3}\cdot3^{3}\cdot19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( \frac{13676826625}{700416} a + \frac{201556775125}{700416} \)
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: $\Omega(E/K)$ \( 9.9153867679137740798773190912464473946 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 108 \)  =  \(( 2^{2} \cdot 3 )\cdot3\cdot3\cdot1\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(6\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 3.9399751820749132960245004898835962503 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 3.939975182 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 9.915387 \cdot 1 \cdot 108 } { {6^2 \cdot 7.549834} } \approx 3.939975182$

Local data at primes of bad reduction

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

This elliptic curve is semistable. There are 4 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((a-4)\) \(2\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((a+3)\) \(2\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((4a+13)\) \(3\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((10a-43)\) \(19\) \(1\) \(I_{1}\) 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
\(2\) 2B
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 228.1-h consists of curves linked by isogenies of degrees dividing 12.

Base change

This elliptic curve is a \(\Q\)-curve.

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