Normalized defining polynomial
\( x^{16} - 3 x^{15} + 3 x^{14} - x^{13} + 2 x^{12} - 8 x^{11} + 14 x^{10} - 18 x^{9} + 29 x^{8} - 49 x^{7} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(36005299687890625\) \(\medspace = 5^{8}\cdot 19^{4}\cdot 29^{4}\) | 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: | \(10.83\) | 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: | $5^{1/2}19^{1/2}29^{1/2}\approx 52.48809388804284$ | ||
Ramified primes: | \(5\), \(19\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{12651607}a^{15}+\frac{31251}{12651607}a^{14}+\frac{2545018}{12651607}a^{13}+\frac{1339362}{12651607}a^{12}-\frac{3747613}{12651607}a^{11}+\frac{680896}{12651607}a^{10}+\frac{720624}{12651607}a^{9}+\frac{2522018}{12651607}a^{8}+\frac{3638991}{12651607}a^{7}-\frac{4922265}{12651607}a^{6}+\frac{3070876}{12651607}a^{5}+\frac{2067731}{12651607}a^{4}+\frac{456175}{12651607}a^{3}-\frac{1067672}{12651607}a^{2}+\frac{5918595}{12651607}a+\frac{622177}{12651607}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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)
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Fundamental units: | $\frac{12061026}{12651607}a^{15}-\frac{35507039}{12651607}a^{14}+\frac{34894177}{12651607}a^{13}-\frac{10628075}{12651607}a^{12}+\frac{22207787}{12651607}a^{11}-\frac{94124937}{12651607}a^{10}+\frac{165036220}{12651607}a^{9}-\frac{209949274}{12651607}a^{8}+\frac{338479101}{12651607}a^{7}-\frac{570831561}{12651607}a^{6}+\frac{765940914}{12651607}a^{5}-\frac{813933705}{12651607}a^{4}+\frac{640263740}{12651607}a^{3}-\frac{350888837}{12651607}a^{2}+\frac{166002370}{12651607}a-\frac{43247557}{12651607}$, $\frac{611945}{12651607}a^{15}-\frac{5336589}{12651607}a^{14}+\frac{10869917}{12651607}a^{13}-\frac{5828798}{12651607}a^{12}-\frac{1539609}{12651607}a^{11}-\frac{9773825}{12651607}a^{10}+\frac{35794909}{12651607}a^{9}-\frac{45884527}{12651607}a^{8}+\frac{52999425}{12651607}a^{7}-\frac{98815686}{12651607}a^{6}+\frac{165238966}{12651607}a^{5}-\frac{189949808}{12651607}a^{4}+\frac{160772811}{12651607}a^{3}-\frac{90814595}{12651607}a^{2}+\frac{28474957}{12651607}a-\frac{12008400}{12651607}$, $\frac{12008400}{12651607}a^{15}-\frac{35413255}{12651607}a^{14}+\frac{30688611}{12651607}a^{13}-\frac{1138483}{12651607}a^{12}+\frac{18188002}{12651607}a^{11}-\frac{97606809}{12651607}a^{10}+\frac{158343775}{12651607}a^{9}-\frac{180356291}{12651607}a^{8}+\frac{302359073}{12651607}a^{7}-\frac{535412175}{12651607}a^{6}+\frac{693738714}{12651607}a^{5}-\frac{687357434}{12651607}a^{4}+\frac{494528992}{12651607}a^{3}-\frac{235504389}{12651607}a^{2}+\frac{113328205}{12651607}a-\frac{43575443}{12651607}$, $\frac{3122743}{12651607}a^{15}-\frac{5654905}{12651607}a^{14}+\frac{1265542}{12651607}a^{13}+\frac{1203443}{12651607}a^{12}+\frac{5425397}{12651607}a^{11}-\frac{16461120}{12651607}a^{10}+\frac{20169363}{12651607}a^{9}-\frac{23953733}{12651607}a^{8}+\frac{48826162}{12651607}a^{7}-\frac{75770743}{12651607}a^{6}+\frac{81233113}{12651607}a^{5}-\frac{71416492}{12651607}a^{4}+\frac{22249467}{12651607}a^{3}+\frac{76807}{12651607}a^{2}-\frac{799149}{12651607}a+\frac{16887735}{12651607}$, $\frac{5158034}{12651607}a^{15}-\frac{13055860}{12651607}a^{14}+\frac{7254626}{12651607}a^{13}+\frac{1473923}{12651607}a^{12}+\frac{11804423}{12651607}a^{11}-\frac{39339557}{12651607}a^{10}+\frac{49517865}{12651607}a^{9}-\frac{56666570}{12651607}a^{8}+\frac{115158494}{12651607}a^{7}-\frac{193028336}{12651607}a^{6}+\frac{225098780}{12651607}a^{5}-\frac{219493535}{12651607}a^{4}+\frac{159457767}{12651607}a^{3}-\frac{84330281}{12651607}a^{2}+\frac{62460872}{12651607}a-\frac{9163209}{12651607}$, $\frac{3122743}{12651607}a^{15}-\frac{5654905}{12651607}a^{14}+\frac{1265542}{12651607}a^{13}+\frac{1203443}{12651607}a^{12}+\frac{5425397}{12651607}a^{11}-\frac{16461120}{12651607}a^{10}+\frac{20169363}{12651607}a^{9}-\frac{23953733}{12651607}a^{8}+\frac{48826162}{12651607}a^{7}-\frac{75770743}{12651607}a^{6}+\frac{81233113}{12651607}a^{5}-\frac{71416492}{12651607}a^{4}+\frac{22249467}{12651607}a^{3}+\frac{76807}{12651607}a^{2}-\frac{13450756}{12651607}a+\frac{16887735}{12651607}$, $\frac{611945}{12651607}a^{15}-\frac{5336589}{12651607}a^{14}+\frac{10869917}{12651607}a^{13}-\frac{5828798}{12651607}a^{12}-\frac{1539609}{12651607}a^{11}-\frac{9773825}{12651607}a^{10}+\frac{35794909}{12651607}a^{9}-\frac{45884527}{12651607}a^{8}+\frac{52999425}{12651607}a^{7}-\frac{98815686}{12651607}a^{6}+\frac{165238966}{12651607}a^{5}-\frac{189949808}{12651607}a^{4}+\frac{160772811}{12651607}a^{3}-\frac{90814595}{12651607}a^{2}+\frac{28474957}{12651607}a+\frac{643207}{12651607}$ | 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: | \( 30.5819893814 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 30.5819893814 \cdot 1}{2\cdot\sqrt{36005299687890625}}\cr\approx \mathstrut & 0.195745356895 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 16T388):
A solvable group of order 128 |
The 20 conjugacy class representatives for $C_2\wr D_4$ |
Character table for $C_2\wr D_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.4.725.1, 4.2.475.1, 4.2.13775.1, 8.2.9986875.1 x2, 8.0.6543125.1 x2, 8.4.189750625.1 |
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: | 8.0.6543125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(29\) | 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
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 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |