Properties

Label 2.2.104.1-16.1-b2
Base field \(\Q(\sqrt{26}) \)
Conductor norm \( 16 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{26}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 26 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-26, 0, 1]))
 
gp: K = nfinit(Polrev([-26, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26, 0, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-14a+77\right){x}-277a+1415\)
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([0,0]),K([77,-14]),K([1415,-277])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,1]),Polrev([0,0]),Polrev([77,-14]),Polrev([1415,-277])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,1],K![0,0],K![77,-14],K![1415,-277]]);
 

This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

trivial

Invariants

Conductor: $\frak{N}$ = \((4)\) = \((2,a)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 16 \) = \(2^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $-4096$
Discriminant ideal: $(\Delta)$ = \((-4096)\) = \((2,a)^{24}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\Delta)$ = \( 16777216 \) = \(2^{24}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
Minimal discriminant: $\frak{D}_{\mathrm{min}}$ = \((64)\) = \((2,a)^{12}\)
Minimal discriminant norm: $N(\frak{D}_{\mathrm{min}})$ = \( 4096 \) = \(2^{12}\)
j-invariant: $j$ = \( 64 \)
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: $\Omega(E/K)$ \( 21.289112115898858603301341815485675910 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 1 \)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(1\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 2.0875691943467857742256996067683401680 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 2.087569194 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 21.289112 \cdot 1 \cdot 1 } { {1^2 \cdot 10.198039} } \approx 2.087569194$

Local data at primes of bad reduction

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

This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((2,a)\) \(2\) \(1\) \(II^{*}\) Additive \(1\) \(4\) \(12\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Nn
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 16.1-b consists of curves linked by isogenies of degree 5.

Base change

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

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