Normalized defining polynomial
\( x^{8} - 2x^{7} - 22x^{6} + 46x^{5} + 82x^{4} - 126x^{3} - 55x^{2} + 76x - 16 \)
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: | \(39213900625\) \(\medspace = 5^{4}\cdot 89^{4}\) | 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: | \(21.10\) | 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: | $5^{1/2}89^{1/2}\approx 21.095023109728988$ | ||
Ramified primes: | \(5\), \(89\) | 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 not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{28}a^{6}-\frac{5}{28}a^{5}-\frac{5}{28}a^{4}-\frac{5}{28}a^{3}-\frac{11}{28}a^{2}+\frac{9}{28}a+\frac{2}{7}$, $\frac{1}{224}a^{7}+\frac{3}{224}a^{6}+\frac{25}{224}a^{5}+\frac{11}{224}a^{4}+\frac{89}{224}a^{3}-\frac{65}{224}a^{2}-\frac{15}{56}a-\frac{3}{14}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
| |
Fundamental units: | $\frac{19}{32}a^{7}-\frac{31}{32}a^{6}-\frac{429}{32}a^{5}+\frac{713}{32}a^{4}+\frac{1811}{32}a^{3}-\frac{1675}{32}a^{2}-\frac{411}{8}a+\frac{47}{2}$, $\frac{57}{224}a^{7}-\frac{93}{224}a^{6}-\frac{1287}{224}a^{5}+\frac{2171}{224}a^{4}+\frac{5497}{224}a^{3}-\frac{5505}{224}a^{2}-\frac{199}{8}a+\frac{159}{14}$, $\frac{55}{224}a^{7}-\frac{99}{224}a^{6}-\frac{175}{32}a^{5}+\frac{323}{32}a^{4}+\frac{4871}{224}a^{3}-\frac{5487}{224}a^{2}-\frac{999}{56}a+\frac{137}{14}$, $\frac{53}{112}a^{7}-\frac{85}{112}a^{6}-\frac{1207}{112}a^{5}+\frac{1971}{112}a^{4}+\frac{5265}{112}a^{3}-\frac{4849}{112}a^{2}-\frac{91}{2}a+\frac{139}{7}$, $\frac{1}{112}a^{7}+\frac{3}{112}a^{6}-\frac{31}{112}a^{5}-\frac{45}{112}a^{4}+\frac{313}{112}a^{3}-\frac{9}{112}a^{2}-\frac{169}{28}a+\frac{18}{7}$, $\frac{15}{224}a^{7}-\frac{11}{224}a^{6}-\frac{353}{224}a^{5}+\frac{333}{224}a^{4}+\frac{1839}{224}a^{3}-\frac{1143}{224}a^{2}-\frac{435}{56}a+\frac{39}{14}$, $\frac{369}{224}a^{7}-\frac{549}{224}a^{6}-\frac{8367}{224}a^{5}+\frac{12675}{224}a^{4}+\frac{35969}{224}a^{3}-\frac{27833}{224}a^{2}-\frac{1095}{8}a+\frac{893}{14}$ | 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: | \( 745.653255453 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 745.653255453 \cdot 1}{2\cdot\sqrt{39213900625}}\cr\approx \mathstrut & 0.481977612412 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{445}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{5}, \sqrt{89})\), 4.4.2225.1 x2, 4.4.39605.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.4.39605.1, 4.4.2225.1 |
Minimal sibling: | 4.4.2225.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(89\) | 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.89.2t1.a.a | $1$ | $ 89 $ | \(\Q(\sqrt{89}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.445.2t1.a.a | $1$ | $ 5 \cdot 89 $ | \(\Q(\sqrt{445}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.445.4t3.b.a | $2$ | $ 5 \cdot 89 $ | 8.8.39213900625.1 | $D_4$ (as 8T4) | $1$ | $2$ |