Normalized defining polynomial
\( x^{16} - 20x^{12} - 48x^{10} - 138x^{8} + 384x^{6} + 172x^{4} + 48x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2393397489569403764736\) \(\medspace = 2^{52}\cdot 3^{12}\) | 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: | \(21.69\) | 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: | $2^{13/4}3^{3/4}\approx 21.686448086636275$ | ||
Ramified primes: | \(2\), \(3\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{3}{8}a^{2}-\frac{1}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{3880}a^{12}-\frac{231}{3880}a^{10}-\frac{61}{1940}a^{8}+\frac{201}{1940}a^{6}-\frac{159}{3880}a^{4}+\frac{417}{3880}a^{2}-\frac{147}{970}$, $\frac{1}{7760}a^{13}-\frac{1}{7760}a^{12}+\frac{127}{3880}a^{11}-\frac{127}{3880}a^{10}-\frac{607}{7760}a^{9}+\frac{607}{7760}a^{8}-\frac{71}{970}a^{7}+\frac{71}{970}a^{6}+\frac{811}{7760}a^{5}-\frac{811}{7760}a^{4}+\frac{1421}{3880}a^{3}-\frac{1421}{3880}a^{2}-\frac{3013}{7760}a+\frac{3013}{7760}$, $\frac{1}{7760}a^{14}-\frac{1}{7760}a^{12}-\frac{387}{7760}a^{10}-\frac{13}{7760}a^{8}+\frac{151}{7760}a^{6}-\frac{263}{7760}a^{4}+\frac{747}{7760}a^{2}+\frac{209}{1552}$, $\frac{1}{7760}a^{15}-\frac{1}{7760}a^{12}-\frac{133}{7760}a^{11}-\frac{127}{3880}a^{10}-\frac{31}{388}a^{9}+\frac{607}{7760}a^{8}-\frac{417}{7760}a^{7}+\frac{71}{970}a^{6}+\frac{137}{1940}a^{5}-\frac{811}{7760}a^{4}+\frac{37}{80}a^{3}-\frac{1421}{3880}a^{2}-\frac{123}{485}a+\frac{3013}{7760}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | 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: | $\frac{23}{485}a^{15}+\frac{19}{1940}a^{13}-\frac{733}{776}a^{11}-\frac{9591}{3880}a^{9}-\frac{13751}{1940}a^{7}+\frac{1618}{97}a^{5}+\frac{44267}{3880}a^{3}+\frac{19549}{3880}a$, $a$, $\frac{119}{776}a^{15}-\frac{9}{1940}a^{13}-\frac{11927}{3880}a^{11}-\frac{14087}{1940}a^{9}-\frac{80751}{3880}a^{7}+\frac{115961}{1940}a^{5}+\frac{97739}{3880}a^{3}+\frac{7147}{1940}a$, $\frac{177}{7760}a^{14}-\frac{11}{1552}a^{12}-\frac{3561}{7760}a^{10}-\frac{1497}{1552}a^{8}-\frac{21229}{7760}a^{6}+\frac{77611}{7760}a^{4}+\frac{16253}{7760}a^{2}-\frac{1231}{7760}$, $\frac{69}{970}a^{15}+\frac{19}{1940}a^{14}+\frac{47}{7760}a^{13}+\frac{7}{7760}a^{12}-\frac{5519}{3880}a^{11}-\frac{77}{388}a^{10}-\frac{27389}{7760}a^{9}-\frac{3839}{7760}a^{8}-\frac{980}{97}a^{7}-\frac{653}{485}a^{6}+\frac{204813}{7760}a^{5}+\frac{5973}{1552}a^{4}+\frac{54371}{3880}a^{3}+\frac{1498}{485}a^{2}+\frac{23353}{7760}a+\frac{4911}{7760}$, $\frac{69}{970}a^{15}-\frac{19}{1940}a^{14}+\frac{47}{7760}a^{13}-\frac{7}{7760}a^{12}-\frac{5519}{3880}a^{11}+\frac{77}{388}a^{10}-\frac{27389}{7760}a^{9}+\frac{3839}{7760}a^{8}-\frac{980}{97}a^{7}+\frac{653}{485}a^{6}+\frac{204813}{7760}a^{5}-\frac{5973}{1552}a^{4}+\frac{54371}{3880}a^{3}-\frac{1498}{485}a^{2}+\frac{23353}{7760}a-\frac{4911}{7760}$, $\frac{19}{3880}a^{15}+\frac{319}{3880}a^{14}+\frac{219}{7760}a^{13}-\frac{83}{7760}a^{12}-\frac{419}{3880}a^{11}-\frac{801}{485}a^{10}-\frac{6143}{7760}a^{9}-\frac{28959}{7760}a^{8}-\frac{7069}{3880}a^{7}-\frac{41551}{3880}a^{6}-\frac{13027}{7760}a^{5}+\frac{259031}{7760}a^{4}+\frac{9821}{776}a^{3}+\frac{5421}{485}a^{2}-\frac{2281}{7760}a+\frac{1047}{1552}$, $\frac{403}{7760}a^{15}+\frac{19}{1940}a^{14}-\frac{13}{776}a^{13}+\frac{7}{7760}a^{12}-\frac{8049}{7760}a^{11}-\frac{77}{388}a^{10}-\frac{209}{97}a^{9}-\frac{3839}{7760}a^{8}-\frac{49591}{7760}a^{7}-\frac{653}{485}a^{6}+\frac{85837}{3880}a^{5}+\frac{5973}{1552}a^{4}+\frac{18637}{7760}a^{3}+\frac{1013}{485}a^{2}+\frac{627}{970}a+\frac{4911}{7760}$, $\frac{7}{80}a^{15}+\frac{311}{7760}a^{14}+\frac{13}{776}a^{13}+\frac{27}{3880}a^{12}-\frac{13637}{7760}a^{11}-\frac{6307}{7760}a^{10}-\frac{1763}{388}a^{9}-\frac{8039}{3880}a^{8}-\frac{98963}{7760}a^{7}-\frac{43959}{7760}a^{6}+\frac{123331}{3880}a^{5}+\frac{58611}{3880}a^{4}+\frac{180601}{7760}a^{3}+\frac{92067}{7760}a^{2}+\frac{12997}{1940}a+\frac{1289}{776}$ (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: | \( 53618.6042833 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 53618.6042833 \cdot 1}{2\cdot\sqrt{2393397489569403764736}}\cr\approx \mathstrut & 0.539482406936 \end{aligned}\] (assuming GRH)
Galois group
$C_4^2:C_2$ (as 16T30):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4^2:C_2$ |
Character table for $C_4^2:C_2$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.2.18432.2, \(\Q(\sqrt{2}, \sqrt{3})\), 4.2.18432.1, 8.4.1358954496.3, 8.4.764411904.3, 8.4.12230590464.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 sibling: | 16.0.2393397489569403764736.1 |
Minimal sibling: | 16.0.2393397489569403764736.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.52.13 | $x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 12 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{5} + 30$ | $16$ | $1$ | $52$ | 16T30 | $[2, 3, 3, 4]^{2}$ |
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |