Base field 4.4.11525.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 11 x^{2} + 5 x + 25 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
Not computed ($ 0 \le r \le 1 $)
Invariants
| Conductor: | $\frak{N}$ | = | \((-1/5a^3-4/5a^2+6/5a+4)\) | = | \((-1/5a^3-4/5a^2+6/5a+4)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 11 \) | = | \(11\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $-1/5a^3-4/5a^2+6/5a+4$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-1/5a^3-4/5a^2+6/5a+4)\) | = | \((-1/5a^3-4/5a^2+6/5a+4)\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 11 \) | = | \(11\) |
|
| |||||
| j-invariant: | $j$ | = | \( -\frac{97445251312609023426791}{11} a^{3} + \frac{250327500830266243970126}{11} a^{2} + \frac{679157934354108520318016}{11} a - \frac{1552759890918669626778665}{11} \) | ||
|
| |||||
| 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: | $\Omega(E/K)$ | ≈ | \( 1.9036052709669598418907280099224628745 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.63866421443728 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | not available |
Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/5a^3-4/5a^2+6/5a+4)\) | \(11\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
11.1-c
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.