Normalized defining polynomial
\( x^{16} - 4x^{14} + 28x^{12} - 106x^{10} + 288x^{8} - 492x^{6} + 657x^{4} - 552x^{2} + 196 \)
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: | \(20882706457600000000\) \(\medspace = 2^{16}\cdot 5^{8}\cdot 13^{8}\) | 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: | \(16.12\) | 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: | $2\cdot 5^{1/2}13^{1/2}\approx 16.1245154965971$ | ||
Ramified primes: | \(2\), \(5\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{3}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{12}+\frac{3}{8}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{56}a^{13}+\frac{1}{56}a^{11}-\frac{1}{28}a^{9}-\frac{11}{56}a^{7}+\frac{9}{56}a^{5}+\frac{11}{28}a^{3}-\frac{1}{2}a^{2}-\frac{3}{7}a$, $\frac{1}{2098376}a^{14}-\frac{69817}{2098376}a^{12}-\frac{65451}{1049188}a^{10}-\frac{28557}{2098376}a^{8}-\frac{331665}{2098376}a^{6}-\frac{1}{2}a^{5}+\frac{130239}{1049188}a^{4}-\frac{242919}{1049188}a^{2}-\frac{1}{2}a-\frac{4421}{10706}$, $\frac{1}{4196752}a^{15}-\frac{1}{4196752}a^{14}+\frac{5125}{4196752}a^{13}+\frac{69817}{4196752}a^{12}+\frac{206337}{4196752}a^{11}-\frac{131395}{4196752}a^{10}-\frac{178441}{4196752}a^{9}+\frac{28557}{4196752}a^{8}+\frac{417755}{4196752}a^{7}+\frac{856259}{4196752}a^{6}+\frac{1721847}{4196752}a^{5}-\frac{1047369}{4196752}a^{4}-\frac{25681}{262297}a^{3}+\frac{514905}{1049188}a^{2}-\frac{20241}{149884}a-\frac{6285}{21412}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{6495}{299768} a^{15} - \frac{19491}{299768} a^{13} + \frac{23391}{42824} a^{11} - \frac{531565}{299768} a^{9} + \frac{1371159}{299768} a^{7} - \frac{1986297}{299768} a^{5} + \frac{1290447}{149884} a^{3} - \frac{169331}{37471} a \) (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: | $\frac{19}{2828}a^{15}-\frac{45}{1414}a^{13}+\frac{155}{808}a^{11}-\frac{4567}{5656}a^{9}+\frac{5815}{2828}a^{7}-\frac{19935}{5656}a^{5}+\frac{21459}{5656}a^{3}-\frac{7975}{2828}a+\frac{1}{2}$, $\frac{7541}{1049188}a^{15}+\frac{580}{262297}a^{14}-\frac{22447}{1049188}a^{13}-\frac{14085}{2098376}a^{12}+\frac{191397}{1049188}a^{11}+\frac{23643}{524594}a^{10}-\frac{150509}{262297}a^{9}-\frac{38349}{262297}a^{8}+\frac{1602325}{1049188}a^{7}+\frac{495493}{2098376}a^{6}-\frac{2101069}{1049188}a^{5}+\frac{238969}{1049188}a^{4}+\frac{2720579}{1049188}a^{3}-\frac{578545}{1049188}a^{2}-\frac{34057}{37471}a+\frac{4609}{10706}$, $\frac{7541}{1049188}a^{15}-\frac{580}{262297}a^{14}-\frac{22447}{1049188}a^{13}+\frac{14085}{2098376}a^{12}+\frac{191397}{1049188}a^{11}-\frac{23643}{524594}a^{10}-\frac{150509}{262297}a^{9}+\frac{38349}{262297}a^{8}+\frac{1602325}{1049188}a^{7}-\frac{495493}{2098376}a^{6}-\frac{2101069}{1049188}a^{5}-\frac{238969}{1049188}a^{4}+\frac{2720579}{1049188}a^{3}+\frac{578545}{1049188}a^{2}-\frac{34057}{37471}a-\frac{4609}{10706}$, $\frac{4901}{2098376}a^{15}-\frac{5147}{2098376}a^{14}-\frac{3177}{262297}a^{13}+\frac{1209}{2098376}a^{12}+\frac{140173}{2098376}a^{11}-\frac{88399}{2098376}a^{10}-\frac{640679}{2098376}a^{9}+\frac{96559}{2098376}a^{8}+\frac{403267}{524594}a^{7}+\frac{50879}{2098376}a^{6}-\frac{2919559}{2098376}a^{5}-\frac{604893}{2098376}a^{4}+\frac{1021749}{524594}a^{3}+\frac{114722}{262297}a^{2}-\frac{159935}{74942}a-\frac{6069}{10706}$, $\frac{75629}{4196752}a^{15}+\frac{927}{85648}a^{14}-\frac{226225}{4196752}a^{13}-\frac{2589}{85648}a^{12}+\frac{1911747}{4196752}a^{11}+\frac{22415}{85648}a^{10}-\frac{6129099}{4196752}a^{9}-\frac{71343}{85648}a^{8}+\frac{16007413}{4196752}a^{7}+\frac{172749}{85648}a^{6}-\frac{22308355}{4196752}a^{5}-\frac{240903}{85648}a^{4}+\frac{14474287}{2098376}a^{3}+\frac{34479}{10706}a^{2}-\frac{137458}{37471}a-\frac{34759}{21412}$, $\frac{233}{524594}a^{15}-\frac{2033}{262297}a^{14}+\frac{17683}{2098376}a^{13}+\frac{19975}{2098376}a^{12}+\frac{614}{262297}a^{11}-\frac{168459}{1049188}a^{10}+\frac{326525}{2098376}a^{9}+\frac{88744}{262297}a^{8}-\frac{538095}{2098376}a^{7}-\frac{1512027}{2098376}a^{6}+\frac{250651}{524594}a^{5}+\frac{314035}{524594}a^{4}-\frac{38923}{2098376}a^{3}-\frac{1718877}{1049188}a^{2}+\frac{68309}{149884}a+\frac{824}{5353}$, $\frac{233}{524594}a^{15}+\frac{2033}{262297}a^{14}+\frac{17683}{2098376}a^{13}-\frac{19975}{2098376}a^{12}+\frac{614}{262297}a^{11}+\frac{168459}{1049188}a^{10}+\frac{326525}{2098376}a^{9}-\frac{88744}{262297}a^{8}-\frac{538095}{2098376}a^{7}+\frac{1512027}{2098376}a^{6}+\frac{250651}{524594}a^{5}-\frac{314035}{524594}a^{4}-\frac{38923}{2098376}a^{3}+\frac{1718877}{1049188}a^{2}+\frac{68309}{149884}a-\frac{824}{5353}$ | 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: | \( 2108.57432833 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2108.57432833 \cdot 2}{4\cdot\sqrt{20882706457600000000}}\cr\approx \mathstrut & 0.560408190583 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 16T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.0.27040000.2, 8.4.285610000.1, 8.0.4569760000.2, 8.0.182790400.2 |
Minimal sibling: | 8.0.27040000.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(5\) | 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |