This is the quartic field with Galois group $A_4$ with the smallest absolute discriminant.
Normalized defining polynomial
\( x^{4} - 2x^{3} + 2x^{2} + 2 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(0, 2)$ |
| |
| Discriminant: |
\(3136\)
\(\medspace = 2^{6}\cdot 7^{2}\)
|
| |
| Root discriminant: | \(7.48\) |
| |
| Galois root discriminant: | $2^{3/2}7^{2/3}\approx 10.350079527967589$ | ||
| Ramified primes: |
\(2\), \(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$a^{3}-a^{2}-1$
|
| |
| Regulator: | \( 2.89814944536 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 2.89814944536 \cdot 1}{2\cdot\sqrt{3136}}\cr\approx \mathstrut & 1.02155673289 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 4 conjugacy class representatives for $A_4$ |
| Character table for $A_4$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | 12.0.1511207993344.1 |
| Degree 6 sibling: | 6.2.153664.1 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.6a2.1 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $$[2, 2]^{3}$$ |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *12 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| *12 | 3.3136.4t4.a.a | $3$ | $ 2^{6} \cdot 7^{2}$ | 4.0.3136.1 | $A_4$ (as 4T4) | $1$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.