Normalized defining polynomial
\( x^{8} - x^{7} - x^{5} + x^{4} + 2x^{3} - 2x + 1 \)
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: | \(6543125\) \(\medspace = 5^{4}\cdot 19^{2}\cdot 29\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(7.11\) | 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: | $5^{1/2}19^{1/2}29^{1/2}\approx 52.48809388804284$ | ||
Ramified primes: | \(5\), \(19\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$
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: | \( -1 \) (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: | $\frac{2}{5}a^{7}-\frac{3}{5}a^{6}-\frac{1}{5}a^{5}-\frac{4}{5}a^{4}+\frac{4}{5}a^{3}+\frac{7}{5}a^{2}-\frac{1}{5}a-\frac{6}{5}$, $a$, $\frac{4}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{3}{5}a^{4}-\frac{2}{5}a^{3}+\frac{9}{5}a^{2}+\frac{8}{5}a-\frac{7}{5}$ | 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: | \( 1.00719516812 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1.00719516812 \cdot 1}{2\cdot\sqrt{6543125}}\cr\approx \mathstrut & 0.306838960532 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
A solvable group of order 128 |
The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.2.475.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 32 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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(29\) | 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.2755.2t1.a.a | $1$ | $ 5 \cdot 19 \cdot 29 $ | \(\Q(\sqrt{-2755}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.95.2t1.a.a | $1$ | $ 5 \cdot 19 $ | \(\Q(\sqrt{-95}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.551.2t1.a.a | $1$ | $ 19 \cdot 29 $ | \(\Q(\sqrt{-551}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.2755.4t3.c.a | $2$ | $ 5 \cdot 19 \cdot 29 $ | 4.2.13775.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.2755.4t3.d.a | $2$ | $ 5 \cdot 19 \cdot 29 $ | 4.2.13775.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.95.4t3.c.a | $2$ | $ 5 \cdot 19 $ | 4.2.475.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.79895.4t3.b.a | $2$ | $ 5 \cdot 19 \cdot 29^{2}$ | 4.2.399475.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.52345.4t3.a.a | $2$ | $ 5 \cdot 19^{2} \cdot 29 $ | 4.0.1518005.3 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
2.145.4t3.a.a | $2$ | $ 5 \cdot 29 $ | 4.4.4205.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
4.1518005.8t29.e.a | $4$ | $ 5 \cdot 19^{2} \cdot 29^{2}$ | 8.4.159580275625.4 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ | |
4.4972775.8t35.g.a | $4$ | $ 5^{2} \cdot 19^{3} \cdot 29 $ | 8.0.6543125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $2$ | |
4.11584775.8t35.g.a | $4$ | $ 5^{2} \cdot 19 \cdot 29^{3}$ | 8.0.6543125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $-2$ | |
4.37950125.8t29.e.a | $4$ | $ 5^{3} \cdot 19^{2} \cdot 29^{2}$ | 8.4.159580275625.4 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ | |
4.4182103775.8t35.g.a | $4$ | $ 5^{2} \cdot 19^{3} \cdot 29^{3}$ | 8.0.6543125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $2$ | |
* | 4.13775.8t35.g.a | $4$ | $ 5^{2} \cdot 19 \cdot 29 $ | 8.0.6543125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $-2$ |