Normalized defining polynomial
\( x^{4} - x^{3} - 7x^{2} + 7x - 2 \)
Invariants
Degree: | $4$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-13356\) \(\medspace = -\,2^{2}\cdot 3^{2}\cdot 7\cdot 53\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.75\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{2/3}3^{1/2}7^{1/2}53^{1/2}\approx 52.95832566335522$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-371}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
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
Trivial group, which has order $1$
Unit group
Rank: | $2$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $a^{3}-a^{2}-8a+5$, $7a^{3}-4a^{2}-51a+27$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 11.2441574258 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 11.2441574258 \cdot 1}{2\cdot\sqrt{13356}}\cr\approx \mathstrut & 1.22263993278 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | deg 24 |
Degree 6 siblings: | 6.2.19820304.1, 6.0.7353332784.2 |
Degree 8 sibling: | 8.0.24552778165776.12 |
Degree 12 siblings: | deg 12, deg 12 |
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 | R | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(53\) | 53.2.1.2 | $x^{2} + 106$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.371.2t1.a.a | $1$ | $ 7 \cdot 53 $ | \(\Q(\sqrt{-371}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.1484.3t2.a.a | $2$ | $ 2^{2} \cdot 7 \cdot 53 $ | 3.1.1484.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
3.4955076.6t8.a.a | $3$ | $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 53^{2}$ | 4.2.13356.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.13356.4t5.a.a | $3$ | $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $ | 4.2.13356.1 | $S_4$ (as 4T5) | $1$ | $1$ |