Base field \(\Q(\sqrt{-39}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 10 \); class number \(4\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((3,a+1)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-6 : 0 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((12)\) | = | \((2,a)^{2}\cdot(2,a+1)^{2}\cdot(3,a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 144 \) | = | \(2^{2}\cdot2^{2}\cdot3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||||
| Discriminant: | $\Delta$ | = | $5038848$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((5038848)\) | = | \((2,a)^{8}\cdot(2,a+1)^{8}\cdot(3,a+1)^{18}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||||
| Discriminant norm: | $N(\Delta)$ | = | \( 25389989167104 \) | = | \(2^{8}\cdot2^{8}\cdot3^{18}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||||
| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((6912)\) | = | \((2,a)^{8}\cdot(2,a+1)^{8}\cdot(3,a+1)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 47775744 \) | = | \(2^{8}\cdot2^{8}\cdot3^{6}\) |
| j-invariant: | $j$ | = | \( 54000 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-3}]\) (potential complex multiplication) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | ||
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: | $\Omega(E/K)$ | ≈ | \( 5.1081157178325565351221945057517738028 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(1\cdot1\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.6359062786385493640155244333506133810 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$\displaystyle 1.635906279 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 4 \cdot 5.108116 \cdot 1 \cdot 2 } { {2^2 \cdot 6.244998} } \approx 1.635906279$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \((2,a+1)\) | \(2\) | \(1\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \((3,a+1)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
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.1.1 |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
144.3-a
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 |
|---|---|
| \(\Q\) | 36.a1 |
| \(\Q\) | 6084.i2 |