Normalized defining polynomial
\( x^{16} - 6 x^{15} + 20 x^{14} - 54 x^{13} + 126 x^{12} - 228 x^{11} + 322 x^{10} - 426 x^{9} + 591 x^{8} + \cdots + 9 \)
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: | \(531312517142544384\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 13^{6}\) | 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: | \(12.82\) | 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: | $2^{3/2}3^{1/2}13^{1/2}\approx 17.663521732655695$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{12}+\frac{1}{6}a^{10}-\frac{1}{6}a^{8}-\frac{1}{6}a^{6}+\frac{1}{6}a^{4}+\frac{1}{6}a^{2}$, $\frac{1}{6}a^{13}+\frac{1}{6}a^{11}-\frac{1}{6}a^{9}-\frac{1}{6}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}$, $\frac{1}{6}a^{14}+\frac{1}{6}a^{10}-\frac{1}{6}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{21838914}a^{15}-\frac{206009}{7279638}a^{14}-\frac{72245}{949518}a^{13}+\frac{450299}{7279638}a^{12}+\frac{128069}{3639819}a^{11}-\frac{254059}{3639819}a^{10}-\frac{2144188}{10919457}a^{9}+\frac{552999}{2426546}a^{8}+\frac{388139}{3639819}a^{7}-\frac{634936}{3639819}a^{6}-\frac{4394396}{10919457}a^{5}+\frac{3574525}{7279638}a^{4}+\frac{9729673}{21838914}a^{3}-\frac{930805}{2426546}a^{2}-\frac{2963675}{7279638}a+\frac{311511}{1213273}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | \( -\frac{17360959}{10919457} a^{15} + \frac{20457609}{2426546} a^{14} - \frac{24641123}{949518} a^{13} + \frac{164723255}{2426546} a^{12} - \frac{1115942683}{7279638} a^{11} + \frac{933187892}{3639819} a^{10} - \frac{3655498930}{10919457} a^{9} + \frac{3244813889}{7279638} a^{8} - \frac{1530480337}{2426546} a^{7} + \frac{2578306688}{3639819} a^{6} - \frac{6769219775}{10919457} a^{5} + \frac{4295053055}{7279638} a^{4} - \frac{13055213141}{21838914} a^{3} + \frac{2892637961}{7279638} a^{2} - \frac{973814915}{7279638} a + \frac{23292528}{1213273} \) (order $12$) | 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: | $\frac{9094334}{10919457}a^{15}-\frac{30982465}{7279638}a^{14}+\frac{12041563}{949518}a^{13}-\frac{39819324}{1213273}a^{12}+\frac{533503067}{7279638}a^{11}-\frac{287213545}{2426546}a^{10}+\frac{1628679071}{10919457}a^{9}-\frac{733181171}{3639819}a^{8}+\frac{698676407}{2426546}a^{7}-\frac{2241621409}{7279638}a^{6}+\frac{2812265788}{10919457}a^{5}-\frac{934739654}{3639819}a^{4}+\frac{5668572799}{21838914}a^{3}-\frac{182937719}{1213273}a^{2}+\frac{291361435}{7279638}a-\frac{5046468}{1213273}$, $\frac{16434227}{21838914}a^{15}-\frac{4515593}{1213273}a^{14}+\frac{5236894}{474759}a^{13}-\frac{207684755}{7279638}a^{12}+\frac{153677069}{2426546}a^{11}-\frac{245625895}{2426546}a^{10}+\frac{2788076081}{21838914}a^{9}-\frac{210548830}{1213273}a^{8}+\frac{1796527609}{7279638}a^{7}-\frac{1905054041}{7279638}a^{6}+\frac{4813690045}{21838914}a^{5}-\frac{805089173}{3639819}a^{4}+\frac{2413263703}{10919457}a^{3}-\frac{923980699}{7279638}a^{2}+\frac{260897669}{7279638}a-\frac{14441935}{2426546}$, $a-1$, $\frac{4645519}{21838914}a^{15}-\frac{8393639}{7279638}a^{14}+\frac{3327667}{949518}a^{13}-\frac{33046211}{3639819}a^{12}+\frac{24789273}{1213273}a^{11}-\frac{122875984}{3639819}a^{10}+\frac{929943745}{21838914}a^{9}-\frac{135789473}{2426546}a^{8}+\frac{292615711}{3639819}a^{7}-\frac{323793007}{3639819}a^{6}+\frac{1595268275}{21838914}a^{5}-\frac{502496243}{7279638}a^{4}+\frac{1593489439}{21838914}a^{3}-\frac{110507009}{2426546}a^{2}+\frac{32636162}{3639819}a+\frac{3863283}{2426546}$, $\frac{20395255}{10919457}a^{15}-\frac{72457133}{7279638}a^{14}+\frac{14589322}{474759}a^{13}-\frac{97621023}{1213273}a^{12}+\frac{1324806655}{7279638}a^{11}-\frac{740891041}{2426546}a^{10}+\frac{4366812760}{10919457}a^{9}-\frac{1937166271}{3639819}a^{8}+\frac{1826066885}{2426546}a^{7}-\frac{6170775185}{7279638}a^{6}+\frac{8152291121}{10919457}a^{5}-\frac{2584819933}{3639819}a^{4}+\frac{15656609717}{21838914}a^{3}-\frac{582719787}{1213273}a^{2}+\frac{616778116}{3639819}a-\frac{32042598}{1213273}$, $\frac{18031963}{21838914}a^{15}-\frac{15555052}{3639819}a^{14}+\frac{6203840}{474759}a^{13}-\frac{82923081}{2426546}a^{12}+\frac{559587169}{7279638}a^{11}-\frac{464209952}{3639819}a^{10}+\frac{3636997885}{21838914}a^{9}-\frac{812783594}{3639819}a^{8}+\frac{2293539529}{7279638}a^{7}-\frac{425720863}{1213273}a^{6}+\frac{6733054235}{21838914}a^{5}-\frac{1078329974}{3639819}a^{4}+\frac{3250552307}{10919457}a^{3}-\frac{712461997}{3639819}a^{2}+\frac{495176869}{7279638}a-\frac{26876231}{2426546}$, $\frac{11300921}{10919457}a^{15}-\frac{20737334}{3639819}a^{14}+\frac{17137081}{949518}a^{13}-\frac{57801699}{1213273}a^{12}+\frac{395651794}{3639819}a^{11}-\frac{226838748}{1213273}a^{10}+\frac{2738133689}{10919457}a^{9}-\frac{1203985100}{3639819}a^{8}+\frac{563695239}{1213273}a^{7}-\frac{1964576888}{3639819}a^{6}+\frac{5340025333}{10919457}a^{5}-\frac{1650080279}{3639819}a^{4}+\frac{4994018459}{10919457}a^{3}-\frac{399782068}{1213273}a^{2}+\frac{942194797}{7279638}a-\frac{25782857}{1213273}$ | 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: | \( 1024.23596776 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1024.23596776 \cdot 1}{12\cdot\sqrt{531312517142544384}}\cr\approx \mathstrut & 0.284434780004 \end{aligned}\]
Galois group
$C_2^2\wr C_2$ (as 16T46):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
Character table for $C_2^2\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.4.7488.1, 4.0.7488.1, 4.0.832.1, 4.0.7488.4, 4.0.117.1, \(\Q(\zeta_{12})\), 4.0.1872.1, 8.0.3504384.2, 8.0.56070144.2, 8.0.56070144.5 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $4$ | $4$ | $24$ | |||
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |