Normalized defining polynomial
\( x^{8} - 3x^{7} + 10x^{6} - 15x^{5} + 23x^{4} - 15x^{3} + 10x^{2} - x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(253605625\) \(\medspace = 5^{4}\cdot 7^{4}\cdot 13^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(11.23\) | 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: | $5^{1/2}7^{1/2}13^{1/2}\approx 21.330729007701542$ | ||
Ramified primes: | \(5\), \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{11}a^{7}+\frac{1}{11}a^{6}+\frac{3}{11}a^{5}-\frac{3}{11}a^{4}-\frac{4}{11}a^{2}+\frac{5}{11}a-\frac{3}{11}$
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)
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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)
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Fundamental units: | $\frac{6}{11}a^{7}-\frac{16}{11}a^{6}+\frac{51}{11}a^{5}-\frac{62}{11}a^{4}+8a^{3}-\frac{24}{11}a^{2}+\frac{8}{11}a+\frac{4}{11}$, $\frac{4}{11}a^{7}-\frac{18}{11}a^{6}+\frac{56}{11}a^{5}-\frac{111}{11}a^{4}+14a^{3}-\frac{148}{11}a^{2}+\frac{64}{11}a-\frac{12}{11}$, $\frac{1}{11}a^{7}+\frac{1}{11}a^{6}+\frac{3}{11}a^{5}+\frac{8}{11}a^{4}+a^{3}+\frac{7}{11}a^{2}+\frac{27}{11}a-\frac{3}{11}$ | 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: | \( 6.41003720329 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 6.41003720329 \cdot 1}{2\cdot\sqrt{253605625}}\cr\approx \mathstrut & 0.313668268743 \end{aligned}\]
Galois group
A solvable group of order 16 |
The 10 conjugacy class representatives for $Q_8:C_2$ |
Character table for $Q_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}, \sqrt{-7})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.1836923935996687890625.1 |
Degree 8 siblings: | 8.4.874680625.1, 8.0.1714374025.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(7\) | 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(13\) | 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.455.2t1.a.a | $1$ | $ 5 \cdot 7 \cdot 13 $ | \(\Q(\sqrt{-455}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.91.2t1.a.a | $1$ | $ 7 \cdot 13 $ | \(\Q(\sqrt{-91}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.65.2t1.a.a | $1$ | $ 5 \cdot 13 $ | \(\Q(\sqrt{65}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.455.8t11.a.a | $2$ | $ 5 \cdot 7 \cdot 13 $ | 8.0.253605625.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
* | 2.455.8t11.a.b | $2$ | $ 5 \cdot 7 \cdot 13 $ | 8.0.253605625.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |