Normalized defining polynomial
\( x^{8} - 2x^{7} - 15x^{6} + 28x^{5} + 44x^{4} - 58x^{3} - 20x^{2} + 12x + 1 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(5184000000\) \(\medspace = 2^{12}\cdot 3^{4}\cdot 5^{6}\) | 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: | \(16.38\) | 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))
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Galois root discriminant: | $2^{3/2}3^{1/2}5^{3/4}\approx 16.3807251762544$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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 and abelian over $\Q$. | |||
Conductor: | \(120=2^{3}\cdot 3\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{120}(1,·)$, $\chi_{120}(49,·)$, $\chi_{120}(113,·)$, $\chi_{120}(109,·)$, $\chi_{120}(77,·)$, $\chi_{120}(17,·)$, $\chi_{120}(53,·)$, $\chi_{120}(61,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{4983}a^{7}-\frac{28}{453}a^{6}+\frac{1217}{4983}a^{5}+\frac{1351}{4983}a^{4}-\frac{478}{1661}a^{3}-\frac{43}{151}a^{2}+\frac{778}{1661}a+\frac{1699}{4983}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{12}{151}a^{7}-\frac{65}{453}a^{6}-\frac{194}{151}a^{5}+\frac{920}{453}a^{4}+\frac{2132}{453}a^{3}-\frac{2009}{453}a^{2}-\frac{1744}{453}a+\frac{613}{453}$, $\frac{232}{1661}a^{7}-\frac{160}{453}a^{6}-\frac{3348}{1661}a^{5}+\frac{25085}{4983}a^{4}+\frac{25112}{4983}a^{3}-\frac{5224}{453}a^{2}-\frac{3316}{4983}a+\frac{14821}{4983}$, $\frac{890}{4983}a^{7}-\frac{52}{151}a^{6}-\frac{13130}{4983}a^{5}+\frac{23080}{4983}a^{4}+\frac{35930}{4983}a^{3}-\frac{3523}{453}a^{2}-\frac{8956}{4983}a+\frac{1861}{1661}$, $\frac{100}{1661}a^{7}-\frac{95}{453}a^{6}-\frac{1214}{1661}a^{5}+\frac{14965}{4983}a^{4}+\frac{1660}{4983}a^{3}-\frac{3215}{453}a^{2}+\frac{10885}{4983}a+\frac{13061}{4983}$, $\frac{232}{1661}a^{7}-\frac{160}{453}a^{6}-\frac{3348}{1661}a^{5}+\frac{25085}{4983}a^{4}+\frac{25112}{4983}a^{3}-\frac{5224}{453}a^{2}-\frac{8299}{4983}a+\frac{14821}{4983}$, $\frac{1799}{4983}a^{7}-\frac{80}{151}a^{6}-\frac{28052}{4983}a^{5}+\frac{35287}{4983}a^{4}+\frac{92783}{4983}a^{3}-\frac{4816}{453}a^{2}-\frac{49972}{4983}a-\frac{1574}{1661}$, $\frac{7}{453}a^{7}-\frac{14}{151}a^{6}-\frac{88}{453}a^{5}+\frac{548}{453}a^{4}+\frac{79}{453}a^{3}-\frac{1024}{453}a^{2}+\frac{181}{453}a+\frac{139}{151}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 130.424947215 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 130.424947215 \cdot 1}{2\cdot\sqrt{5184000000}}\cr\approx \mathstrut & 0.231866572827 \end{aligned}\]
Galois group
$C_2\times C_4$ (as 8T2):
An abelian group of order 8 |
The 8 conjugacy class representatives for $C_4\times C_2$ |
Character table for $C_4\times C_2$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.72000.1, \(\Q(\zeta_{15})^+\) |
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 | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |