Base field \(\Q(\sqrt{-129}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 129 \); class number \(12\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((23,a+3)\). No global minimal model exists.
Mordell-Weil group structure
Not computed ($ 0 \le r \le 1 $)
Mordell-Weil generators
No non-torsion generators are known.
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{11}{2} a + 124 : -\frac{239}{4} a - \frac{1669}{4} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((12,2a+6)\) | = | \((2,a+1)^{3}\cdot(3,a)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 24 \) | = | \(2^{3}\cdot3\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $-160392960a-42595398432$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-160392960a-42595398432)\) | = | \((2,a+1)^{10}\cdot(3,a)^{4}\cdot(23,a+3)^{12}\) |
|
| |||||
| Discriminant norm: | $N(\Delta)$ | = | \( 1817686608889493505024 \) | = | \(2^{10}\cdot3^{4}\cdot23^{12}\) |
|
| |||||
| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((288)\) | = | \((2,a+1)^{10}\cdot(3,a)^{4}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 82944 \) | = | \(2^{10}\cdot3^{4}\) |
| j-invariant: | $j$ | = | \( \frac{3065617154}{9} \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
|
|
|||
| Mordell-Weil rank: | $r?$ | \(0 \le r \le 1\) | |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | not available |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | not available |
| Global period: | $\Omega(E/K)$ | ≈ | \( 7.2706940358628777611249866685015359950 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2\cdot2^{2}\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 11.391523780018055385465124390601343280 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | not available |
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a+1)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(-1\) | \(3\) | \(10\) | \(0\) |
| \((3,a)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((23,a+3)\) | \(23\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
24.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 144.b1 |
| \(\Q\) | 44376.i1 |