Base field \(\Q(\sqrt{13}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 3 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$\left(4 : -4 a - 2 : 1\right)$ | $0.39521495450573230182313238877914535268$ | $\infty$ |
$\left(-\frac{5}{4} a + \frac{9}{4} : -a + \frac{11}{8} : 1\right)$ | $0$ | $2$ |
Invariants
Conductor: | $\frak{N}$ | = | \((33)\) | = | \((-a)\cdot(-a+1)\cdot(11)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||||
Conductor norm: | $N(\frak{N})$ | = | \( 1089 \) | = | \(3\cdot3\cdot121\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||||
Discriminant: | $\Delta$ | = | $363a$ | ||
Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((363a)\) | = | \((-a)^{2}\cdot(-a+1)\cdot(11)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||||
Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 395307 \) | = | \(3^{2}\cdot3\cdot121^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||||
j-invariant: | $j$ | = | \( -\frac{829562110871}{1089} a + \frac{1910298012317}{1089} \) | ||
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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.39521495450573230182313238877914535268 \) |
Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.790429909011464603646264777558290705360 \) |
Global period: | $\Omega(E/K)$ | ≈ | \( 16.566701771284791719815078047603576828 \) |
Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot1\cdot2\) |
Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.6318486614814730400843515395421251438 \) |
Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$\displaystyle 3.631848661 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 16.566702 \cdot 0.790430 \cdot 4 } { {2^2 \cdot 3.605551} } \approx 3.631848661$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $\frak{p}$ of bad reduction.
$\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((-a+1)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((11)\) | \(121\) | \(2\) | \(I_{2}\) | 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 |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
1089.1-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.