Normalized defining polynomial
\( x^{8} - 4x^{7} + 9x^{6} - 12x^{5} + 7x^{4} + 12x^{3} - 15x^{2} - 2x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1244678400\) \(\medspace = 2^{8}\cdot 3^{4}\cdot 5^{2}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.71\) | 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: | $2\cdot 3^{1/2}5^{1/2}7^{1/2}\approx 20.493901531919196$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{333}a^{7}+\frac{25}{333}a^{6}+\frac{68}{333}a^{5}-\frac{38}{333}a^{4}-\frac{32}{111}a^{3}-\frac{12}{37}a^{2}-\frac{50}{111}a-\frac{23}{333}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{91}{333}a^{7}-\frac{389}{333}a^{6}+\frac{860}{333}a^{5}-\frac{1127}{333}a^{4}+\frac{196}{111}a^{3}+\frac{129}{37}a^{2}-\frac{554}{111}a-\frac{428}{333}$, $a$, $\frac{80}{333}a^{7}-\frac{331}{333}a^{6}+\frac{778}{333}a^{5}-\frac{1042}{333}a^{4}+\frac{215}{111}a^{3}+\frac{113}{37}a^{2}-\frac{448}{111}a-\frac{175}{333}$, $\frac{35}{111}a^{7}-\frac{124}{111}a^{6}+\frac{271}{111}a^{5}-\frac{331}{111}a^{4}+\frac{64}{37}a^{3}+\frac{146}{37}a^{2}-\frac{85}{37}a-\frac{28}{111}$, $\frac{196}{333}a^{7}-\frac{761}{333}a^{6}+\frac{1673}{333}a^{5}-\frac{2120}{333}a^{4}+\frac{388}{111}a^{3}+\frac{275}{37}a^{2}-\frac{809}{111}a-\frac{845}{333}$ | 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: | \( 51.3613526633 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 51.3613526633 \cdot 1}{2\cdot\sqrt{1244678400}}\cr\approx \mathstrut & 0.459787965797 \end{aligned}\]
Galois group
$D_4:C_2^2$ (as 8T22):
A solvable group of order 32 |
The 17 conjugacy class representatives for $Q_8:C_2^2$ |
Character table for $Q_8:C_2^2$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{3}, \sqrt{7})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 siblings: | data not computed |
Degree 16 siblings: | 16.0.403275801600000000.1, 16.0.1549224319426560000.1, 16.0.968265199641600000000.2, 16.0.11953891353600000000.1, 16.0.3782285936100000000.3, 16.8.968265199641600000000.2, 16.0.968265199641600000000.9, 16.0.968265199641600000000.11, 16.0.968265199641600000000.7 |
Minimal sibling: | 8.0.77792400.1 |
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 | R | R | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(5\) | 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.140.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | \(\Q(\sqrt{35}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.420.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 \cdot 7 $ | \(\Q(\sqrt{-105}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.105.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 7 $ | \(\Q(\sqrt{105}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.60.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{15}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.28.2t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\sqrt{7}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.84.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 7 $ | \(\Q(\sqrt{-21}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 4.176400.8t22.j.a | $4$ | $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ | 8.4.1244678400.1 | $Q_8:C_2^2$ (as 8T22) | $1$ | $0$ |