Normalized defining polynomial
\( x^{8} - 4x^{7} + 14x^{6} - 12x^{5} + 12x^{4} + 32x^{3} + 50x^{2} - 32x + 101 \)
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: | \(2837708800\) \(\medspace = 2^{18}\cdot 5^{2}\cdot 433\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.19\) | 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: | $2^{9/4}5^{1/2}433^{1/2}\approx 221.33313760167954$ | ||
Ramified primes: | \(2\), \(5\), \(433\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{433}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{27}a^{6}+\frac{4}{27}a^{5}-\frac{2}{9}a^{4}+\frac{2}{27}a^{3}-\frac{11}{27}a^{2}+\frac{2}{27}a-\frac{10}{27}$, $\frac{1}{2349}a^{7}-\frac{13}{783}a^{6}+\frac{335}{2349}a^{5}+\frac{530}{2349}a^{4}-\frac{529}{2349}a^{3}+\frac{16}{2349}a^{2}-\frac{257}{783}a-\frac{974}{2349}$
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: | \( \frac{2}{783} a^{7} + \frac{1}{87} a^{6} - \frac{26}{783} a^{5} + \frac{16}{783} a^{4} + \frac{160}{783} a^{3} - \frac{403}{783} a^{2} - \frac{12}{29} a + \frac{53}{783} \) (order $4$) | 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: | $\frac{35}{2349}a^{7}-\frac{26}{261}a^{6}+\frac{589}{2349}a^{5}-\frac{764}{2349}a^{4}-\frac{593}{2349}a^{3}+\frac{647}{2349}a^{2}+\frac{37}{261}a-\frac{4684}{2349}$, $\frac{19}{783}a^{7}-\frac{16}{783}a^{6}+\frac{130}{783}a^{5}-\frac{22}{783}a^{4}+\frac{178}{261}a^{3}+\frac{140}{261}a^{2}-\frac{149}{783}a+\frac{1127}{783}$, $\frac{143}{2349}a^{7}+\frac{287}{783}a^{6}-\frac{728}{2349}a^{5}+\frac{5842}{2349}a^{4}+\frac{9265}{2349}a^{3}+\frac{7421}{2349}a^{2}+\frac{1732}{783}a+\frac{17927}{2349}$ | 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: | \( 61.7568130411 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 61.7568130411 \cdot 1}{4\cdot\sqrt{2837708800}}\cr\approx \mathstrut & 0.451710769305 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), 4.0.320.1 |
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 | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.18.61 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{3} + 2$ | $8$ | $1$ | $18$ | $C_2^3 : C_4 $ | $[2, 2, 3]^{4}$ |
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(433\) | $\Q_{433}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{433}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |