Normalized defining polynomial
\( x^{8} - 3x^{7} + 72x^{6} - 592x^{5} + 3519x^{4} - 9968x^{3} + 23522x^{2} - 66393x + 114383 \)
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)
| |
Discriminant: | \(2871098559212689\) \(\medspace = 13^{6}\cdot 29^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(85.56\) | 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: | $13^{3/4}29^{3/4}\approx 85.55709155460606$ | ||
Ramified primes: | \(13\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 8.0.2871098559212689.1$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{561773331271622}a^{7}-\frac{56003192436472}{280886665635811}a^{6}-\frac{238817984582889}{561773331271622}a^{5}+\frac{10027963035559}{561773331271622}a^{4}+\frac{172971772845169}{561773331271622}a^{3}-\frac{63473188827392}{280886665635811}a^{2}+\frac{39289559222951}{561773331271622}a-\frac{99072546782413}{280886665635811}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{6}\times C_{6}$, which has order $72$
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)
| |
Fundamental units: | $\frac{1201834}{55086618089}a^{7}-\frac{17239}{55086618089}a^{6}+\frac{66267139}{55086618089}a^{5}-\frac{561532900}{55086618089}a^{4}+\frac{1448900099}{55086618089}a^{3}-\frac{3339334251}{55086618089}a^{2}+\frac{11836222877}{55086618089}a-\frac{35551622663}{55086618089}$, $\frac{366282}{4532842733}a^{7}-\frac{996197}{4532842733}a^{6}+\frac{24026452}{4532842733}a^{5}-\frac{214585483}{4532842733}a^{4}+\frac{1044207604}{4532842733}a^{3}-\frac{2778508982}{4532842733}a^{2}+\frac{3585491830}{4532842733}a-\frac{1734221451}{4532842733}$, $\frac{54020677380}{280886665635811}a^{7}-\frac{246397948030}{280886665635811}a^{6}+\frac{3928011753770}{280886665635811}a^{5}-\frac{36009336957644}{280886665635811}a^{4}+\frac{214498000759466}{280886665635811}a^{3}-\frac{586539872275282}{280886665635811}a^{2}+\frac{526611286219502}{280886665635811}a+\frac{376921675499543}{280886665635811}$ | 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: | \( 193.47300399 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 193.47300399 \cdot 72}{2\cdot\sqrt{2871098559212689}}\cr\approx \mathstrut & 0.20259013714 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{13}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}, \sqrt{29})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.4.3.1 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
13.4.3.1 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(29\) | 29.4.3.1 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.377.2t1.a.a | $1$ | $ 13 \cdot 29 $ | \(\Q(\sqrt{377}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.142129.8t5.a.a | $2$ | $ 13^{2} \cdot 29^{2}$ | 8.0.2871098559212689.1 | $Q_8$ (as 8T5) | $-1$ | $-2$ |