Normalized defining polynomial
\( x^{8} - 2x^{7} + 35x^{6} + 226x^{5} - 1636x^{4} - 2730x^{3} + 26315x^{2} + 41850x + 21425 \)
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: | \(48641067427840000\) \(\medspace = 2^{14}\cdot 5^{4}\cdot 41^{6}\) | 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: | \(121.86\) | 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: | $2^{2}5^{1/2}41^{3/4}\approx 144.92152186193653$ | ||
Ramified primes: | \(2\), \(5\), \(41\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{480}a^{6}+\frac{17}{120}a^{5}-\frac{1}{3}a^{4}+\frac{43}{240}a^{3}-\frac{1}{5}a^{2}-\frac{11}{24}a-\frac{25}{96}$, $\frac{1}{163337962080}a^{7}-\frac{49474757}{81668981040}a^{6}+\frac{9979594213}{20417245260}a^{5}+\frac{36435668203}{81668981040}a^{4}+\frac{1352780441}{13611496840}a^{3}+\frac{202095757}{1408085880}a^{2}-\frac{4829704849}{32667592416}a+\frac{346018501}{5444598736}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (assuming GRH)
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: | $\frac{71541487}{27222993680}a^{7}-\frac{172774369}{10208622630}a^{6}+\frac{168464362}{1020862263}a^{5}-\frac{5179831097}{40834490520}a^{4}-\frac{39510504287}{10208622630}a^{3}+\frac{477419019}{46936196}a^{2}+\frac{416162985787}{16333796208}a+\frac{3474673031}{4083449052}$, $\frac{2908183}{6805748420}a^{7}-\frac{92725183}{20417245260}a^{6}-\frac{15526447}{1020862263}a^{5}+\frac{399765737}{10208622630}a^{4}+\frac{7726347743}{10208622630}a^{3}-\frac{25630793}{11734049}a^{2}-\frac{17874569449}{4083449052}a-\frac{10257039077}{4083449052}$, $\frac{3427722983}{10208622630}a^{7}-\frac{11938476877}{10208622630}a^{6}+\frac{22319798502}{1701437105}a^{5}+\frac{96962166853}{1701437105}a^{4}-\frac{3267158720417}{5104311315}a^{3}-\frac{663819308}{176010735}a^{2}+\frac{18922173738005}{2041724526}a+\frac{399959377715}{2041724526}$ (assuming GRH) | 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: | \( 7710.20684948 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 7710.20684948 \cdot 32}{2\cdot\sqrt{48641067427840000}}\cr\approx \mathstrut & 0.871774049584 \end{aligned}\] (assuming GRH)
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
\(\Q(\sqrt{-82}) \), 4.0.4410944.1 |
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: | 8.0.7233948160000.67 |
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 | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.4 | $x^{4} + 6 x^{2} + 4 x + 6$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(5\) | 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(41\) | 41.4.3.2 | $x^{4} + 82$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
41.4.3.2 | $x^{4} + 82$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |