Normalized defining polynomial
\( x^{8} - 2x^{7} - 2x^{6} - 4x^{5} + 40x^{4} - 14x^{3} - 37x^{2} - 142x + 236 \)
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: | \(11051265625\) \(\medspace = 5^{6}\cdot 29^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.01\) | 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: | $5^{3/4}29^{3/4}\approx 41.78553833475025$ | ||
Ramified primes: | \(5\), \(29\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{10}a-\frac{2}{5}$, $\frac{1}{10}a^{5}-\frac{1}{2}a^{2}-\frac{2}{5}$, $\frac{1}{10}a^{6}-\frac{1}{2}a^{3}-\frac{2}{5}a$, $\frac{1}{340}a^{7}-\frac{3}{68}a^{6}-\frac{11}{340}a^{5}+\frac{3}{340}a^{4}+\frac{137}{340}a^{3}+\frac{109}{340}a^{2}+\frac{21}{170}a-\frac{2}{85}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{4}$, which has order $4$
Unit group
Rank: | $3$ | 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: | $\frac{3}{340}a^{7}-\frac{11}{340}a^{6}+\frac{1}{340}a^{5}-\frac{5}{68}a^{4}+\frac{21}{68}a^{3}-\frac{47}{340}a^{2}+\frac{6}{85}a-\frac{176}{85}$, $\frac{7}{340}a^{7}-\frac{3}{340}a^{6}-\frac{43}{340}a^{5}+\frac{21}{340}a^{4}+\frac{109}{340}a^{3}+\frac{253}{340}a^{2}-\frac{227}{170}a-\frac{48}{85}$, $\frac{3}{340}a^{7}-\frac{11}{340}a^{6}+\frac{7}{68}a^{5}-\frac{59}{340}a^{4}-\frac{31}{340}a^{3}-\frac{81}{340}a^{2}+\frac{369}{170}a-\frac{176}{85}$ | 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: | \( 26.2355904197 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 26.2355904197 \cdot 4}{2\cdot\sqrt{11051265625}}\cr\approx \mathstrut & 0.777918863227 \end{aligned}\]
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
\(\Q(\sqrt{5}) \), 4.0.3625.1 |
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: | 16.8.86380562306022715087890625.2, 16.0.102711726879931884765625.6, 16.0.3455222492240908603515625.6, 16.0.86380562306022715087890625.8 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.4.3.1 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.145.4t1.d.a | $1$ | $ 5 \cdot 29 $ | 4.4.3048625.2 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.145.4t1.c.a | $1$ | $ 5 \cdot 29 $ | 4.4.3048625.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.145.4t1.c.b | $1$ | $ 5 \cdot 29 $ | 4.4.3048625.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.145.4t1.d.b | $1$ | $ 5 \cdot 29 $ | 4.4.3048625.2 | $C_4$ (as 4T1) | $0$ | $1$ | |
2.4205.4t3.b.a | $2$ | $ 5 \cdot 29^{2}$ | 4.0.121945.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
* | 2.725.4t3.a.a | $2$ | $ 5^{2} \cdot 29 $ | 4.0.3625.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ |
* | 4.3048625.8t19.b.a | $4$ | $ 5^{3} \cdot 29^{3}$ | 8.0.11051265625.3 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |