Properties

Label 2.2.17.1-612.1-e5
Base field \(\Q(\sqrt{17}) \)
Conductor norm \( 612 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 16 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{17}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-4, -1, 1]))
 
gp: K = nfinit(Polrev([-4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}-34{x}+68\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-34,0]),K([68,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-34,0]),Polrev([68,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-34,0],K![68,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z/{2}\Z \oplus \Z/{8}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(\frac{9}{4} a - \frac{13}{4} : -\frac{9}{8} a + \frac{13}{8} : 1\right)$$0$$2$
$\left(-4 : 14 : 1\right)$$0$$8$

Invariants

Conductor: $\frak{N}$ = \((-12a+6)\) = \((-a+2)\cdot(-a-1)\cdot(3)\cdot(-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 612 \) = \(2\cdot2\cdot9\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $352512$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((352512)\) = \((-a+2)^{8}\cdot(-a-1)^{8}\cdot(3)^{4}\cdot(-2a+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 124264710144 \) = \(2^{8}\cdot2^{8}\cdot9^{4}\cdot17^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( \frac{4354703137}{352512} \)
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: $\Omega(E/K)$ \( 8.7577870800086443037951892851946268343 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 512 \)  =  \(2^{3}\cdot2^{3}\cdot2^{2}\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(16\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 4.2481507267700359953716626563758926612 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 4.248150727 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 8.757787 \cdot 1 \cdot 512 } { {16^2 \cdot 4.123106} } \approx 4.248150727$

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 \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-a+2)\) \(2\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((-a-1)\) \(2\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((3)\) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-2a+1)\) \(17\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 612.1-e consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 102.c5
\(\Q\) 1734.j5