Normalized defining polynomial
\( x^{8} - 6x^{6} - 14x^{5} + 46x^{4} + 30x^{3} - 29x^{2} - 172x + 157 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(709689600\) \(\medspace = 2^{8}\cdot 3^{4}\cdot 5^{2}\cdot 37^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.78\) | 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: | $2\cdot 3^{1/2}5^{1/2}37^{1/2}\approx 47.11687595755899$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(37\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{589655}a^{7}+\frac{16042}{589655}a^{6}-\frac{215546}{589655}a^{5}-\frac{287888}{589655}a^{4}+\frac{114572}{589655}a^{3}-\frac{108512}{589655}a^{2}+\frac{29958}{589655}a-\frac{100692}{589655}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{4683}{589655} a^{7} + \frac{2639}{589655} a^{6} - \frac{30489}{589655} a^{5} - \frac{110243}{589655} a^{4} + \frac{190488}{589655} a^{3} + \frac{474707}{589655} a^{2} + \frac{191286}{589655} a - \frac{293534}{117931} \) (order $12$) | 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: | $\frac{4683}{589655}a^{7}+\frac{2639}{589655}a^{6}-\frac{30489}{589655}a^{5}-\frac{110243}{589655}a^{4}+\frac{190488}{589655}a^{3}+\frac{474707}{589655}a^{2}+\frac{191286}{589655}a-\frac{175603}{117931}$, $\frac{5866}{589655}a^{7}-\frac{6566}{589655}a^{6}-\frac{10917}{117931}a^{5}-\frac{97019}{589655}a^{4}+\frac{45289}{117931}a^{3}-\frac{11557}{117931}a^{2}-\frac{219424}{589655}a+\frac{57107}{589655}$, $\frac{11269}{589655}a^{7}-\frac{2185}{117931}a^{6}-\frac{80998}{589655}a^{5}-\frac{45993}{589655}a^{4}+\frac{710866}{589655}a^{3}-\frac{231051}{589655}a^{2}-\frac{338157}{117931}a+\frac{1449106}{589655}$ | 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: | \( 52.8616254518 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 52.8616254518 \cdot 1}{12\cdot\sqrt{709689600}}\cr\approx \mathstrut & 0.257717862154 \end{aligned}\]
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
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\) |
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: | data not computed |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(5\) | 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.555.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 37 $ | \(\Q(\sqrt{-555}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.185.2t1.a.a | $1$ | $ 5 \cdot 37 $ | \(\Q(\sqrt{185}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.2220.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 \cdot 37 $ | \(\Q(\sqrt{555}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.740.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 37 $ | \(\Q(\sqrt{-185}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.444.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 37 $ | \(\Q(\sqrt{111}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.148.2t1.a.a | $1$ | $ 2^{2} \cdot 37 $ | \(\Q(\sqrt{-37}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.60.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{15}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.111.2t1.a.a | $1$ | $ 3 \cdot 37 $ | \(\Q(\sqrt{-111}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.37.2t1.a.a | $1$ | $ 37 $ | \(\Q(\sqrt{37}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 4.4928400.8t22.f.a | $4$ | $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}$ | 8.0.709689600.1 | $Q_8:C_2^2$ (as 8T22) | $1$ | $0$ |