Normalized defining polynomial
\( x^{16} - 8x^{14} + 16x^{12} + 82x^{8} - 168x^{6} + 144x^{4} + 81 \)
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: | \(1494186269970473680896\) \(\medspace = 2^{64}\cdot 3^{4}\) | 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: | \(21.06\) | 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: | $2^{35/8}3^{1/2}\approx 35.93907196672871$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}+\frac{1}{4}$, $\frac{1}{4}a^{9}+\frac{1}{4}a$, $\frac{1}{24}a^{10}+\frac{1}{24}a^{8}-\frac{1}{2}a^{7}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{24}a^{2}-\frac{1}{2}a+\frac{3}{8}$, $\frac{1}{24}a^{11}+\frac{1}{24}a^{9}-\frac{1}{3}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}+\frac{1}{24}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{72}a^{12}+\frac{1}{72}a^{10}-\frac{1}{36}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}+\frac{1}{72}a^{4}-\frac{1}{2}a^{3}-\frac{5}{24}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{72}a^{13}+\frac{1}{72}a^{11}-\frac{1}{36}a^{9}-\frac{1}{2}a^{6}+\frac{1}{72}a^{5}-\frac{1}{2}a^{4}-\frac{5}{24}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{1037880}a^{14}+\frac{2303}{518940}a^{12}+\frac{3761}{207576}a^{10}+\frac{140}{8649}a^{8}+\frac{495937}{1037880}a^{6}-\frac{3025}{11532}a^{4}+\frac{8727}{38440}a^{2}+\frac{1261}{4805}$, $\frac{1}{1037880}a^{15}+\frac{2303}{518940}a^{13}+\frac{3761}{207576}a^{11}+\frac{140}{8649}a^{9}+\frac{495937}{1037880}a^{7}-\frac{3025}{11532}a^{5}+\frac{8727}{38440}a^{3}+\frac{1261}{4805}a$
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: | \( -\frac{28}{8649} a^{14} + \frac{191}{5766} a^{12} - \frac{97}{961} a^{10} + \frac{490}{8649} a^{8} - \frac{1708}{8649} a^{6} + \frac{22799}{17298} a^{4} - \frac{3319}{2883} a^{2} + \frac{210}{961} \) (order $4$) | 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: | $\frac{617}{345960}a^{14}+\frac{443}{57660}a^{12}+\frac{413}{23064}a^{10}-\frac{631}{8649}a^{8}-\frac{49169}{345960}a^{6}-\frac{8143}{34596}a^{4}-\frac{8877}{38440}a^{2}-\frac{3686}{4805}$, $\frac{479}{129735}a^{14}+\frac{11299}{518940}a^{12}+\frac{356}{25947}a^{10}-\frac{5773}{34596}a^{8}-\frac{52283}{129735}a^{6}+\frac{7381}{34596}a^{4}+\frac{9727}{4805}a^{2}+\frac{21127}{19220}$, $\frac{557}{345960}a^{15}-\frac{1247}{207576}a^{14}+\frac{1711}{115320}a^{13}+\frac{5393}{103788}a^{12}-\frac{1247}{23064}a^{11}-\frac{6365}{51894}a^{10}+\frac{9157}{69192}a^{9}+\frac{43}{7688}a^{8}-\frac{103169}{345960}a^{7}-\frac{64535}{207576}a^{6}+\frac{17749}{69192}a^{5}+\frac{43151}{34596}a^{4}-\frac{176711}{115320}a^{3}-\frac{7121}{5766}a^{2}+\frac{99797}{38440}a-\frac{14661}{7688}$, $\frac{3347}{518940}a^{14}+\frac{18299}{259470}a^{12}-\frac{6709}{25947}a^{10}+\frac{11753}{34596}a^{8}-\frac{331019}{518940}a^{6}+\frac{46787}{17298}a^{4}-\frac{59818}{14415}a^{2}+\frac{48249}{19220}$, $\frac{1}{3348}a^{15}-\frac{53}{23064}a^{14}+\frac{5}{6696}a^{13}-\frac{1}{961}a^{12}-\frac{167}{3348}a^{11}+\frac{907}{11532}a^{10}+\frac{505}{2232}a^{9}-\frac{1475}{23064}a^{8}-\frac{125}{3348}a^{7}-\frac{2359}{7688}a^{6}-\frac{143}{744}a^{5}-\frac{5186}{2883}a^{4}-\frac{1409}{372}a^{3}-\frac{17369}{11532}a^{2}-\frac{67}{248}a-\frac{15625}{7688}$, $\frac{1177}{207576}a^{15}+\frac{77}{38440}a^{14}+\frac{7427}{207576}a^{13}-\frac{3367}{345960}a^{12}-\frac{3577}{207576}a^{11}-\frac{1781}{69192}a^{10}-\frac{821}{7688}a^{9}+\frac{11509}{69192}a^{8}-\frac{152521}{207576}a^{7}+\frac{10247}{115320}a^{6}+\frac{60065}{69192}a^{5}-\frac{40547}{69192}a^{4}+\frac{2311}{23064}a^{3}+\frac{124189}{115320}a^{2}-\frac{8109}{7688}a+\frac{8737}{38440}$, $\frac{7469}{1037880}a^{15}-\frac{1253}{345960}a^{14}+\frac{64051}{1037880}a^{13}+\frac{9097}{345960}a^{12}-\frac{33553}{207576}a^{11}-\frac{1711}{69192}a^{10}+\frac{9835}{69192}a^{9}-\frac{1061}{7688}a^{8}-\frac{651653}{1037880}a^{7}-\frac{64901}{345960}a^{6}+\frac{10981}{7688}a^{5}+\frac{17183}{23064}a^{4}-\frac{328349}{115320}a^{3}+\frac{51797}{38440}a^{2}+\frac{9543}{38440}a-\frac{42977}{38440}$ | 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: | \( 62647.818096813666 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 62647.818096813666 \cdot 1}{4\cdot\sqrt{1494186269970473680896}}\cr\approx \mathstrut & 0.984198128350516 \end{aligned}\]
Galois group
$C_2^4.(C_4\times D_4)$ (as 16T824):
A solvable group of order 512 |
The 41 conjugacy class representatives for $C_2^4.(C_4\times D_4)$ |
Character table for $C_2^4.(C_4\times D_4)$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 4.0.512.1, 8.0.67108864.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.64.848 | $x^{16} + 12 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{6} + 16 x^{5} + 16 x + 18$ | $16$ | $1$ | $64$ | 16T824 | $[2, 3, 3, 7/2, 4, 4, 17/4, 19/4]^{2}$ |
\(3\) | 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |