Normalized defining polynomial
\( x^{8} - 184x^{5} + 590x^{4} - 680x^{3} + 624x^{2} + 720x + 135 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(57420021760000\) \(\medspace = 2^{26}\cdot 5^{4}\cdot 37^{2}\) | 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: | \(52.47\) | 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: | $2^{55/16}5^{1/2}37^{1/2}\approx 147.35893506362828$ | ||
Ramified primes: | \(2\), \(5\), \(37\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{160284771}a^{7}+\frac{3435146}{53428257}a^{6}-\frac{2100907}{17809419}a^{5}-\frac{41092303}{160284771}a^{4}+\frac{699614}{160284771}a^{3}-\frac{47440613}{160284771}a^{2}+\frac{4254530}{17809419}a+\frac{211148}{17809419}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$
Unit group
Rank: | $5$ | 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: | $\frac{15982}{5936473}a^{7}-\frac{3204}{5936473}a^{6}+\frac{39474}{5936473}a^{5}+\frac{2987975}{5936473}a^{4}-\frac{8788762}{5936473}a^{3}+\frac{1418352}{5936473}a^{2}+\frac{4169538}{5936473}a-\frac{110156}{5936473}$, $\frac{1644181}{53428257}a^{7}+\frac{60695}{5936473}a^{6}-\frac{8813}{17809419}a^{5}+\frac{303355303}{53428257}a^{4}-\frac{1068047483}{53428257}a^{3}+\frac{1451745290}{53428257}a^{2}-\frac{170333989}{5936473}a-\frac{89635843}{5936473}$, $\frac{222127}{53428257}a^{7}+\frac{18320}{5936473}a^{6}-\frac{143504}{17809419}a^{5}-\frac{44372020}{53428257}a^{4}+\frac{87215879}{53428257}a^{3}-\frac{25630970}{53428257}a^{2}+\frac{68418266}{17809419}a+\frac{9471569}{5936473}$, $\frac{300283}{53428257}a^{7}-\frac{111437}{5936473}a^{6}-\frac{1050614}{17809419}a^{5}+\frac{46258180}{53428257}a^{4}-\frac{20093657}{53428257}a^{3}+\frac{69048407}{53428257}a^{2}+\frac{13449394}{5936473}a+\frac{3313229}{5936473}$, $\frac{45809914}{53428257}a^{7}-\frac{4432325}{17809419}a^{6}+\frac{1212334}{17809419}a^{5}-\frac{8432210845}{53428257}a^{4}+\frac{29471230649}{53428257}a^{3}-\frac{39663496802}{53428257}a^{2}+\frac{4449333038}{5936473}a+\frac{2382790246}{5936473}$ | 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: | \( 9815.805369923364 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 9815.805369923364 \cdot 8}{2\cdot\sqrt{57420021760000}}\cr\approx \mathstrut & 3.27290932024433 \end{aligned}\]
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
\(\Q(\sqrt{10}) \), 4.2.25600.2 |
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 | ${\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.26.35 | $x^{8} + 4 x^{6} + 8 x^{3} + 10$ | $8$ | $1$ | $26$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 4]^{2}$ |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(37\) | 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |