Normalized defining polynomial
\( x^{8} - 2x^{7} + 4x^{6} - 5x^{5} + 5x^{4} - 5x^{3} + 4x^{2} - 2x + 1 \)
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: | \(2193125\) \(\medspace = 5^{4}\cdot 11^{2}\cdot 29\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(6.20\) | 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: | $5^{1/2}11^{1/2}29^{1/2}\approx 39.93745109543172$ | ||
Ramified primes: | \(5\), \(11\), \(29\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
Monogenic: | Yes | |
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: | $a^{7}-2a^{6}+4a^{5}-5a^{4}+5a^{3}-5a^{2}+4a-2$, $a^{7}-a^{6}+2a^{5}-a^{4}+a^{3}-a^{2}+a$, $a^{7}-2a^{6}+4a^{5}-5a^{4}+5a^{3}-4a^{2}+3a-1$ | 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: | \( 0.454073872008 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 0.454073872008 \cdot 1}{2\cdot\sqrt{2193125}}\cr\approx \mathstrut & 0.2389374903087 \end{aligned}\]
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
\(\Q(\sqrt{5}) \), 4.2.275.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(11\) | 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
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) | $1$ | $1$ | |
* | 1.5.2t1.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) | $1$ | $1$ | |
1.1595.2t1.a.a | $1$ | $ 5 \cdot 11 \cdot 29 $ | \(\Q(\sqrt{-1595}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.55.2t1.a.a | $1$ | $ 5 \cdot 11 $ | \(\Q(\sqrt{-55}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.319.2t1.a.a | $1$ | $ 11 \cdot 29 $ | \(\Q(\sqrt{-319}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.1595.4t3.a.a | $2$ | $ 5 \cdot 11 \cdot 29 $ | 4.0.508805.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.1595.4t3.b.a | $2$ | $ 5 \cdot 11 \cdot 29 $ | 4.0.508805.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.46255.4t3.a.a | $2$ | $ 5 \cdot 11 \cdot 29^{2}$ | 4.0.508805.4 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.55.4t3.b.a | $2$ | $ 5 \cdot 11 $ | 4.0.605.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.145.4t3.a.a | $2$ | $ 5 \cdot 29 $ | 4.4.4205.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
2.17545.4t3.a.a | $2$ | $ 5 \cdot 11^{2} \cdot 29 $ | 4.0.508805.3 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
4.6706975.8t35.f.a | $4$ | $ 5^{2} \cdot 11 \cdot 29^{3}$ | 8.0.2193125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $-2$ | |
4.12720125.8t29.c.a | $4$ | $ 5^{3} \cdot 11^{2} \cdot 29^{2}$ | 8.0.307827025.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ | |
4.964975.8t35.f.a | $4$ | $ 5^{2} \cdot 11^{3} \cdot 29 $ | 8.0.2193125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $2$ | |
4.508805.8t29.c.a | $4$ | $ 5 \cdot 11^{2} \cdot 29^{2}$ | 8.0.307827025.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ | |
4.811543975.8t35.f.a | $4$ | $ 5^{2} \cdot 11^{3} \cdot 29^{3}$ | 8.0.2193125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $2$ | |
* | 4.7975.8t35.f.a | $4$ | $ 5^{2} \cdot 11 \cdot 29 $ | 8.0.2193125.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $-2$ |