Normalized defining polynomial
\( x^{8} - x^{7} + 3x^{6} - 8x^{5} + 9x^{4} - 8x^{3} + 16x^{2} - 20x + 9 \)
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: | \(307827025\) \(\medspace = 5^{2}\cdot 11^{4}\cdot 29^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.51\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{95}a^{7}-\frac{32}{95}a^{6}+\frac{9}{19}a^{5}+\frac{22}{95}a^{4}-\frac{8}{95}a^{3}-\frac{9}{19}a^{2}-\frac{14}{95}a+\frac{34}{95}$
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: | $\frac{54}{95}a^{7}-\frac{18}{95}a^{6}+\frac{30}{19}a^{5}-\frac{332}{95}a^{4}+\frac{233}{95}a^{3}-\frac{49}{19}a^{2}+\frac{669}{95}a-\frac{539}{95}$, $\frac{18}{95}a^{7}-\frac{6}{95}a^{6}+\frac{10}{19}a^{5}-\frac{79}{95}a^{4}+\frac{46}{95}a^{3}-\frac{10}{19}a^{2}+\frac{128}{95}a-\frac{53}{95}$, $\frac{13}{95}a^{7}-\frac{36}{95}a^{6}+\frac{3}{19}a^{5}-\frac{189}{95}a^{4}+\frac{181}{95}a^{3}-\frac{3}{19}a^{2}+\frac{388}{95}a-\frac{413}{95}$ | 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: | \( 12.1412815954 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 12.1412815954 \cdot 1}{2\cdot\sqrt{307827025}}\cr\approx \mathstrut & 0.539263017017 \end{aligned}\]
Galois group
$C_2\wr C_2^2$ (as 8T29):
A solvable group of order 64 |
The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), 4.0.605.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(11\) | 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(29\) | 29.4.2.2 | $x^{4} - 696 x^{2} + 1682$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.1595.2t1.a.a | $1$ | $ 5 \cdot 11 \cdot 29 $ | \(\Q(\sqrt{-1595}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.319.2t1.a.a | $1$ | $ 11 \cdot 29 $ | \(\Q(\sqrt{-319}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.55.2t1.a.a | $1$ | $ 5 \cdot 11 $ | \(\Q(\sqrt{-55}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 2.55.4t3.b.a | $2$ | $ 5 \cdot 11 $ | 4.0.605.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.17545.4t3.b.a | $2$ | $ 5 \cdot 11^{2} \cdot 29 $ | 4.0.87725.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
2.145.4t3.b.a | $2$ | $ 5 \cdot 29 $ | 4.4.725.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
2.1595.4t3.d.a | $2$ | $ 5 \cdot 11 \cdot 29 $ | 4.2.7975.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.46255.4t3.a.a | $2$ | $ 5 \cdot 11 \cdot 29^{2}$ | 4.0.508805.4 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.1595.4t3.c.a | $2$ | $ 5 \cdot 11 \cdot 29 $ | 4.2.7975.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
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) | $1$ | $0$ | |
* | 4.508805.8t29.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) | $1$ | $0$ |