Base field \(\Q(\sqrt{-23}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 6 \); class number \(3\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(2 : -3 a + 12 : 1\right)$ | $0.13693264060765609071667196155970425318$ | $\infty$ |
| $\left(-4 a + 2 : a - 12 : 1\right)$ | $0$ | $2$ |
| $\left(a - 1 : 3 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((18,6a)\) | = | \((2,a)\cdot(2,a+1)\cdot(3,a)^{2}\cdot(3,a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 108 \) | = | \(2\cdot2\cdot3^{2}\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||||
| Discriminant: | $\Delta$ | = | $1229904a-147744$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((1229904a-147744)\) | = | \((2,a)^{20}\cdot(2,a+1)^{4}\cdot(3,a)^{8}\cdot(3,a+2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||||
| Discriminant norm: | $N(\Delta)$ | = | \( 8916100448256 \) | = | \(2^{20}\cdot2^{4}\cdot3^{8}\cdot3^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||||
| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((-14256a+38880)\) | = | \((2,a)^{8}\cdot(2,a+1)^{4}\cdot(3,a)^{8}\cdot(3,a+2)^{4}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 2176782336 \) | = | \(2^{8}\cdot2^{4}\cdot3^{8}\cdot3^{4}\) |
| j-invariant: | $j$ | = | \( \frac{54256267}{20736} a - \frac{3034703}{6912} \) | ||
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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.13693264060765609071667196155970425318 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.273865281215312181433343923119408506360 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 4.2631400829787424910520982906442720882 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 128 \) = \(2^{3}\cdot2\cdot2^{2}\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.9475680945168386727622166971056384619 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$\displaystyle 1.947568095 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 4.263140 \cdot 0.273865 \cdot 128 } { {4^2 \cdot 4.795832} } \approx 1.947568095$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $\frak{p}$ 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((2,a+1)\) | \(2\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((3,a)\) | \(3\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
| \((3,a+2)\) | \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 and 4.
Its isogeny class
108.6-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.