Normalized defining polynomial
\( x^{8} - 3x^{7} - 5x^{6} + 27x^{5} - 23x^{4} - 6x^{3} + 84x^{2} - 360x + 432 \)
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)
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Discriminant: | \(7438545009\) \(\medspace = 3^{4}\cdot 7^{2}\cdot 37^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.14\) | 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: | $3^{1/2}7^{1/2}37^{3/4}\approx 68.74813805509002$ | ||
Ramified primes: | \(3\), \(7\), \(37\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{6}-\frac{1}{4}a^{5}-\frac{5}{12}a^{4}+\frac{1}{4}a^{3}+\frac{1}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{106344}a^{7}-\frac{253}{11816}a^{6}+\frac{2887}{15192}a^{5}+\frac{12847}{35448}a^{4}+\frac{38371}{106344}a^{3}-\frac{17}{4431}a^{2}+\frac{998}{4431}a-\frac{265}{1477}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{223}{15192} a^{7} + \frac{35}{5064} a^{6} + \frac{1619}{15192} a^{5} - \frac{767}{5064} a^{4} + \frac{161}{15192} a^{3} + \frac{61}{844} a^{2} - \frac{1004}{633} a + \frac{648}{211} \) (order $6$) | 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: | $\frac{29}{2954}a^{7}-\frac{181}{8862}a^{6}-\frac{22}{211}a^{5}+\frac{877}{4431}a^{4}+\frac{289}{1477}a^{3}-\frac{1525}{8862}a^{2}+\frac{1895}{2954}a-\frac{3415}{1477}$, $\frac{122}{4431}a^{7}-\frac{118}{4431}a^{6}-\frac{101}{1266}a^{5}+\frac{2899}{8862}a^{4}-\frac{9041}{8862}a^{3}+\frac{8269}{8862}a^{2}+\frac{17657}{2954}a-\frac{12311}{1477}$, $\frac{1017}{5908}a^{7}-\frac{7783}{8862}a^{6}+\frac{429}{422}a^{5}+\frac{11125}{4431}a^{4}-\frac{29795}{2954}a^{3}+\frac{413863}{17724}a^{2}-\frac{60432}{1477}a+\frac{48119}{1477}$ | 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: | \( 198.395438613 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 198.395438613 \cdot 2}{6\cdot\sqrt{7438545009}}\cr\approx \mathstrut & 1.19504952616 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 8T19):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3 : C_4 $ |
Character table for $C_2^3 : C_4 $ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.333.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(37\) | 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.4.3.1 | $x^{4} + 111$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
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) | $1$ | $-1$ |
1.111.2t1.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) | $1$ | $1$ | |
1.777.4t1.a.a | $1$ | $ 3 \cdot 7 \cdot 37 $ | 4.0.22337973.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.259.4t1.a.a | $1$ | $ 7 \cdot 37 $ | 4.4.2481997.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.259.4t1.a.b | $1$ | $ 7 \cdot 37 $ | 4.4.2481997.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.777.4t1.a.b | $1$ | $ 3 \cdot 7 \cdot 37 $ | 4.0.22337973.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 2.111.4t3.b.a | $2$ | $ 3 \cdot 37 $ | 4.2.4107.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.201243.4t3.b.a | $2$ | $ 3 \cdot 7^{2} \cdot 37^{2}$ | 4.0.22337973.3 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.22337973.8t19.b.a | $4$ | $ 3^{2} \cdot 7^{2} \cdot 37^{3}$ | 8.0.7438545009.2 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |