Normalized defining polynomial
\( x^{16} - 12x^{14} + 60x^{12} - 144x^{10} + 191x^{8} - 144x^{6} + 60x^{4} - 12x^{2} + 1 \)
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: | \(1846757322198614016\) \(\medspace = 2^{48}\cdot 3^{8}\) | 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: | \(13.86\) | 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: | $2^{3}3^{1/2}\approx 13.856406460551018$ | ||
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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}+\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{5}a^{12}-\frac{1}{5}$, $\frac{1}{5}a^{13}-\frac{1}{5}a$, $\frac{1}{25}a^{14}+\frac{2}{25}a^{12}-\frac{2}{25}a^{10}+\frac{8}{25}a^{8}+\frac{8}{25}a^{6}-\frac{2}{25}a^{4}+\frac{12}{25}a^{2}+\frac{11}{25}$, $\frac{1}{25}a^{15}+\frac{2}{25}a^{13}-\frac{2}{25}a^{11}+\frac{8}{25}a^{9}+\frac{8}{25}a^{7}-\frac{2}{25}a^{5}+\frac{12}{25}a^{3}+\frac{11}{25}a$
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: | \( -\frac{36}{25} a^{14} + \frac{408}{25} a^{12} - \frac{1888}{25} a^{10} + \frac{3927}{25} a^{8} - \frac{4273}{25} a^{6} + \frac{2387}{25} a^{4} - \frac{617}{25} a^{2} + \frac{39}{25} \) (order $24$) | 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: | $\frac{56}{25}a^{15}-\frac{653}{25}a^{13}+\frac{3138}{25}a^{11}-\frac{7002}{25}a^{9}+\frac{8373}{25}a^{7}-\frac{5462}{25}a^{5}+\frac{1822}{25}a^{3}-\frac{219}{25}a$, $\frac{17}{25}a^{15}-\frac{196}{25}a^{13}+\frac{931}{25}a^{11}-\frac{2049}{25}a^{9}+\frac{2476}{25}a^{7}-\frac{1719}{25}a^{5}+\frac{669}{25}a^{3}-\frac{93}{25}a$, $\frac{8}{25}a^{15}-\frac{74}{25}a^{13}+\frac{229}{25}a^{11}+\frac{9}{25}a^{9}-\frac{841}{25}a^{7}+\frac{1179}{25}a^{5}-\frac{534}{25}a^{3}+\frac{48}{25}a$, $\frac{59}{25}a^{15}-\frac{4}{5}a^{14}-\frac{692}{25}a^{13}+\frac{47}{5}a^{12}+\frac{3352}{25}a^{11}-\frac{229}{5}a^{10}-\frac{7583}{25}a^{9}+\frac{526}{5}a^{8}+\frac{9192}{25}a^{7}-\frac{659}{5}a^{6}-\frac{5948}{25}a^{5}+\frac{441}{5}a^{4}+\frac{1828}{25}a^{3}-28a^{2}-\frac{121}{25}a+\frac{9}{5}$, $\frac{42}{25}a^{15}+\frac{39}{25}a^{14}-\frac{496}{25}a^{13}-\frac{457}{25}a^{12}+\frac{2421}{25}a^{11}+\frac{2207}{25}a^{10}-\frac{5534}{25}a^{9}-\frac{4953}{25}a^{8}+\frac{6716}{25}a^{7}+\frac{5897}{25}a^{6}-\frac{4229}{25}a^{5}-\frac{3743}{25}a^{4}+\frac{1159}{25}a^{3}+\frac{1153}{25}a^{2}-\frac{28}{25}a-\frac{101}{25}$, $\frac{17}{25}a^{15}-\frac{17}{25}a^{14}-\frac{196}{25}a^{13}+\frac{211}{25}a^{12}+\frac{931}{25}a^{11}-\frac{1081}{25}a^{10}-\frac{2049}{25}a^{9}+\frac{2624}{25}a^{8}+\frac{2476}{25}a^{7}-\frac{3226}{25}a^{6}-\frac{1719}{25}a^{5}+\frac{2019}{25}a^{4}+\frac{669}{25}a^{3}-\frac{544}{25}a^{2}-\frac{68}{25}a+\frac{53}{25}$, $\frac{28}{25}a^{15}-\frac{28}{25}a^{14}-\frac{324}{25}a^{13}+\frac{324}{25}a^{12}+\frac{1544}{25}a^{11}-\frac{1544}{25}a^{10}-\frac{3401}{25}a^{9}+\frac{3401}{25}a^{8}+\frac{4024}{25}a^{7}-\frac{4024}{25}a^{6}-\frac{2556}{25}a^{5}+\frac{2556}{25}a^{4}+\frac{836}{25}a^{3}-\frac{836}{25}a^{2}-\frac{87}{25}a+\frac{87}{25}$ | 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: | \( 3719.97240803 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3719.97240803 \cdot 1}{24\cdot\sqrt{1846757322198614016}}\cr\approx \mathstrut & 0.277052777863 \end{aligned}\]
Galois group
A solvable group of order 16 |
The 10 conjugacy class representatives for $Q_8 : C_2$ |
Character table for $Q_8 : C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.4.1358954496.1, 8.0.1358954496.5, 8.0.150994944.1 |
Minimal sibling: | 8.0.150994944.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.48.6 | $x^{16} + 8 x^{15} + 76 x^{14} + 192 x^{13} + 152 x^{12} + 184 x^{11} + 408 x^{10} + 368 x^{9} + 416 x^{8} + 416 x^{7} + 488 x^{6} + 224 x^{5} + 416 x^{4} + 272 x^{3} + 160 x^{2} - 96 x + 228$ | $8$ | $2$ | $48$ | $Q_8 : C_2$ | $[2, 3, 4]^{2}$ |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |