Normalized defining polynomial
\( x^{8} - 3x^{7} - 2x^{6} + 14x^{5} - 15x^{4} - x^{3} + 13x^{2} - 3x + 1 \)
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: | \(2790703125\) \(\medspace = 3^{6}\cdot 5^{7}\cdot 7^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.16\) | 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: | $3^{25/18}5^{7/8}7^{1/2}\approx 49.75264603559782$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{1}{3}a^{3}+\frac{2}{9}a^{2}-\frac{4}{9}a+\frac{1}{9}$
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: | $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{7}{9}a^{5}+a^{4}+a^{3}-\frac{22}{9}a^{2}+\frac{5}{9}a+\frac{4}{9}$, $a$, $\frac{1}{9}a^{7}-\frac{4}{9}a^{6}+\frac{2}{9}a^{5}+\frac{5}{3}a^{4}-\frac{10}{3}a^{3}+\frac{14}{9}a^{2}+\frac{2}{9}a+\frac{4}{9}$, $\frac{2}{3}a^{7}-2a^{6}-a^{5}+\frac{26}{3}a^{4}-\frac{32}{3}a^{3}+2a^{2}+5a-\frac{2}{3}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{7}{9}a^{5}+a^{4}+a^{3}-\frac{31}{9}a^{2}+\frac{23}{9}a+\frac{4}{9}$ | 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: | \( 55.6162254124 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 55.6162254124 \cdot 1}{2\cdot\sqrt{2790703125}}\cr\approx \mathstrut & 0.332502117185 \end{aligned}\]
Galois group
$A_4^2:C_4$ (as 8T46):
A solvable group of order 576 |
The 13 conjugacy class representatives for $A_4^2:C_4$ |
Character table for $A_4^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Degree 36 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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | R | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(5\) | 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(7\) | 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.105.4t1.a.a | $1$ | $ 3 \cdot 5 \cdot 7 $ | 4.0.55125.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.105.4t1.a.b | $1$ | $ 3 \cdot 5 \cdot 7 $ | 4.0.55125.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
4.4465125.6t10.b.a | $4$ | $ 3^{6} \cdot 5^{3} \cdot 7^{2}$ | 6.2.22325625.3 | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ | |
4.4465125.6t10.a.a | $4$ | $ 3^{6} \cdot 5^{3} \cdot 7^{2}$ | 6.2.22325625.3 | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ | |
6.49228003125.12t160.b.a | $6$ | $ 3^{8} \cdot 5^{5} \cdot 7^{4}$ | 8.4.2790703125.1 | $A_4^2:C_4$ (as 8T46) | $1$ | $-2$ | |
* | 6.558140625.8t46.a.a | $6$ | $ 3^{6} \cdot 5^{6} \cdot 7^{2}$ | 8.4.2790703125.1 | $A_4^2:C_4$ (as 8T46) | $1$ | $2$ |
9.498...625.16t1030.a.a | $9$ | $ 3^{12} \cdot 5^{8} \cdot 7^{4}$ | 8.4.2790703125.1 | $A_4^2:C_4$ (as 8T46) | $1$ | $1$ | |
9.996...125.18t184.a.a | $9$ | $ 3^{12} \cdot 5^{7} \cdot 7^{4}$ | 8.4.2790703125.1 | $A_4^2:C_4$ (as 8T46) | $1$ | $1$ | |
9.104...125.36t766.a.a | $9$ | $ 3^{13} \cdot 5^{8} \cdot 7^{5}$ | 8.4.2790703125.1 | $A_4^2:C_4$ (as 8T46) | $0$ | $-1$ | |
9.104...125.36t766.a.b | $9$ | $ 3^{13} \cdot 5^{8} \cdot 7^{5}$ | 8.4.2790703125.1 | $A_4^2:C_4$ (as 8T46) | $0$ | $-1$ | |
12.222...125.24t1505.a.a | $12$ | $ 3^{18} \cdot 5^{11} \cdot 7^{6}$ | 8.4.2790703125.1 | $A_4^2:C_4$ (as 8T46) | $1$ | $0$ |