Properties

Label 2.2.28.1-648.1-m3
Base field \(\Q(\sqrt{7}) \)
Conductor norm \( 648 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{7}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 7 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, 0, 1]))
 
gp: K = nfinit(Polrev([-7, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-466a-1234\right){x}-7779a-20582\)
sage: E = EllipticCurve([K([1,1]),K([1,1]),K([1,1]),K([-1234,-466]),K([-20582,-7779])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([1,1]),Polrev([-1234,-466]),Polrev([-20582,-7779])], K);
 
magma: E := EllipticCurve([K![1,1],K![1,1],K![1,1],K![-1234,-466],K![-20582,-7779]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-8 a - 18 : -a - 4 : 1\right)$$0.64528977739308943704780835994917297593$$\infty$
$\left(-8 a - 21 : 14 a + 38 : 1\right)$$0$$2$
$\left(-\frac{7}{2} a - 9 : \frac{23}{4} a + \frac{65}{4} : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((18a+54)\) = \((a+3)^{3}\cdot(-a+2)^{2}\cdot(-a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 648 \) = \(2^{3}\cdot3^{2}\cdot3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $26244$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((26244)\) = \((a+3)^{4}\cdot(-a+2)^{8}\cdot(-a-2)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 688747536 \) = \(2^{4}\cdot3^{8}\cdot3^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( \frac{35152}{9} \)
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: $\mathrm{Reg}(E/K)$ \( 0.64528977739308943704780835994917297593 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 1.29057955478617887409561671989834595186 \)
Global period: $\Omega(E/K)$ \( 7.5780113566687825347161968372579115861 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 32 \)  =  \(2\cdot2^{2}\cdot2^{2}\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(4\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 3.6965025707268544718535305611532899795 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 3.696502571 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 7.578011 \cdot 1.290580 \cdot 32 } { {4^2 \cdot 5.291503} } \approx 3.696502571$

Local data at primes of bad reduction

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

This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((a+3)\) \(2\) \(2\) \(III\) Additive \(-1\) \(3\) \(4\) \(0\)
\((-a+2)\) \(3\) \(4\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)
\((-a-2)\) \(3\) \(4\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)

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 and 4.
Its isogeny class 648.1-m consists of curves linked by isogenies of degrees dividing 8.

Base change

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

Base field Curve
\(\Q\) 72.a4
\(\Q\) 7056.q4