Properties

Label 2.0.7.1-10404.2-b2
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 10404 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
 
gp: K = nfinit(Polrev([2, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+1809{x}-37790\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([1809,0]),K([-37790,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([1809,0]),Polrev([-37790,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![1809,0],K![-37790,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-\frac{526}{7} : \frac{28611}{49} a - \frac{12489}{49} : 1\right)$$1.7438237887847223921173795565714105505$$\infty$
$\left(\frac{71}{4} : -\frac{75}{8} : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((102)\) = \((a)\cdot(-a+1)\cdot(3)\cdot(17)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 10404 \) = \(2\cdot2\cdot9\cdot289\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $-1001033261568$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-1001033261568)\) = \((a)^{9}\cdot(-a+1)^{9}\cdot(3)^{4}\cdot(17)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 1002067590765467905818624 \) = \(2^{9}\cdot2^{9}\cdot9^{4}\cdot289^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( \frac{655215969476375}{1001033261568} \)
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: $\mathrm{Reg}(E/K)$ \( 1.7438237887847223921173795565714105505 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 3.4876475775694447842347591131428211010 \)
Global period: $\Omega(E/K)$ \( 0.277245289947559450935488653409904975220 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 24 \)  =  \(1\cdot1\cdot2^{2}\cdot( 2 \cdot 3 )\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(2\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 2.1927998897728714468980882752951483524 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 2.192799890 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 0.277245 \cdot 3.487648 \cdot 24 } { {2^2 \cdot 2.645751} } \approx 2.192799890$

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 \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((a)\) \(2\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)
\((-a+1)\) \(2\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)
\((3)\) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((17)\) \(289\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 10404.2-b consists of curves linked by isogenies of degrees dividing 6.

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.b4
\(\Q\) 4998.d4