Normalized defining polynomial
\( x^{8} - 3x^{7} + 44x^{6} - 6x^{5} + 893x^{4} + 780x^{3} + 9188x^{2} + 19953x + 32719 \)
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: | \(6472063200625\) \(\medspace = 5^{4}\cdot 11^{4}\cdot 29^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.94\) | 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: | $5^{1/2}11^{1/2}29^{1/2}\approx 39.93745109543172$ | ||
Ramified primes: | \(5\), \(11\), \(29\) | 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)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{66}a^{6}-\frac{3}{11}a^{5}+\frac{13}{66}a^{4}+\frac{1}{22}a^{3}-\frac{29}{66}a^{2}-\frac{3}{11}a-\frac{29}{66}$, $\frac{1}{375514669662}a^{7}+\frac{214249619}{62585778277}a^{6}-\frac{186416773313}{375514669662}a^{5}-\frac{18991203929}{125171556554}a^{4}+\frac{72903802267}{375514669662}a^{3}+\frac{8914687993}{62585778277}a^{2}+\frac{4193910271}{375514669662}a-\frac{26015102405}{62585778277}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{4}\times C_{4}$, which has order $16$
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: | $\frac{3024}{138771127}a^{7}-\frac{31287}{138771127}a^{6}+\frac{175166}{138771127}a^{5}-\frac{128503}{138771127}a^{4}+\frac{653256}{138771127}a^{3}+\frac{515432}{12615557}a^{2}+\frac{1424512}{12615557}a+\frac{99215043}{138771127}$, $\frac{7065075}{11379232414}a^{7}-\frac{19558745}{5689616207}a^{6}+\frac{288822837}{11379232414}a^{5}-\frac{82620127}{11379232414}a^{4}+\frac{1029712137}{11379232414}a^{3}+\frac{3903914771}{5689616207}a^{2}+\frac{10772638125}{11379232414}a+\frac{747410265}{5689616207}$, $\frac{4006328}{4579447191}a^{7}+\frac{11215277}{9158894382}a^{6}+\frac{106779776}{4579447191}a^{5}+\frac{850709441}{9158894382}a^{4}+\frac{6037635337}{9158894382}a^{3}+\frac{18610363247}{9158894382}a^{2}+\frac{15218049392}{4579447191}a+\frac{19992962519}{9158894382}$ | 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: | \( 116.618863594 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 116.618863594 \cdot 16}{2\cdot\sqrt{6472063200625}}\cr\approx \mathstrut & 0.571553502821 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.0.508805.4 x2, 4.2.231275.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.2.231275.1, 4.0.508805.4 |
Minimal sibling: | 4.2.231275.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(29\) | 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |