Normalized defining polynomial
\( x^{16} - 2 x^{15} + x^{14} + 4 x^{13} - 11 x^{12} + 24 x^{11} - 23 x^{10} - 12 x^{9} + 86 x^{8} - 176 x^{7} + 233 x^{6} - 228 x^{5} + 167 x^{4} - 90 x^{3} + 34 x^{2} - 8 x + 1 \)
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: | \(99038527051792384\) \(\medspace = 2^{34}\cdot 7^{8}\) | 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: | \(11.54\) | 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: | $2^{11/4}7^{1/2}\approx 17.798422345016238$ | ||
Ramified primes: | \(2\), \(7\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6271661}a^{15}+\frac{2892771}{6271661}a^{14}+\frac{3091548}{6271661}a^{13}+\frac{2591387}{6271661}a^{12}-\frac{2539025}{6271661}a^{11}+\frac{762392}{6271661}a^{10}-\frac{2597657}{6271661}a^{9}+\frac{2495904}{6271661}a^{8}-\frac{2961186}{6271661}a^{7}+\frac{2649659}{6271661}a^{6}+\frac{1696778}{6271661}a^{5}-\frac{2734925}{6271661}a^{4}+\frac{116043}{570151}a^{3}-\frac{402448}{6271661}a^{2}-\frac{1091823}{6271661}a-\frac{158909}{6271661}$
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: | \( -1 \) (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: | $\frac{2477069}{6271661}a^{15}-\frac{17647419}{6271661}a^{14}+\frac{9950389}{6271661}a^{13}+\frac{18186186}{6271661}a^{12}-\frac{74565298}{6271661}a^{11}+\frac{125548592}{6271661}a^{10}-\frac{227740993}{6271661}a^{9}-\frac{39358458}{6271661}a^{8}+\frac{498441562}{6271661}a^{7}-\frac{1140584246}{6271661}a^{6}+\frac{1568948189}{6271661}a^{5}-\frac{1505094553}{6271661}a^{4}+\frac{98396081}{570151}a^{3}-\frac{518953503}{6271661}a^{2}+\frac{164257768}{6271661}a-\frac{19113361}{6271661}$, $\frac{667798}{570151}a^{15}-\frac{70244}{570151}a^{14}-\frac{883320}{570151}a^{13}+\frac{2451079}{570151}a^{12}-\frac{2002882}{570151}a^{11}+\frac{6377064}{570151}a^{10}+\frac{6196670}{570151}a^{9}-\frac{18023555}{570151}a^{8}+\frac{29657858}{570151}a^{7}-\frac{27217769}{570151}a^{6}+\frac{10803786}{570151}a^{5}+\frac{8348737}{570151}a^{4}-\frac{18046666}{570151}a^{3}+\frac{14071594}{570151}a^{2}-\frac{5836350}{570151}a+\frac{812644}{570151}$, $\frac{17063384}{6271661}a^{15}-\frac{24608997}{6271661}a^{14}+\frac{6686961}{6271661}a^{13}+\frac{68857581}{6271661}a^{12}-\frac{150845548}{6271661}a^{11}+\frac{339566633}{6271661}a^{10}-\frac{224801787}{6271661}a^{9}-\frac{280812005}{6271661}a^{8}+\frac{1300775038}{6271661}a^{7}-\frac{2356169379}{6271661}a^{6}+\frac{2881475667}{6271661}a^{5}-\frac{2599937531}{6271661}a^{4}+\frac{155390419}{570151}a^{3}-\frac{809954656}{6271661}a^{2}+\frac{237994382}{6271661}a-\frac{35371316}{6271661}$, $\frac{7005150}{6271661}a^{15}-\frac{98179}{6271661}a^{14}-\frac{9201815}{6271661}a^{13}+\frac{26646617}{6271661}a^{12}-\frac{21075902}{6271661}a^{11}+\frac{62480894}{6271661}a^{10}+\frac{78123103}{6271661}a^{9}-\frac{192953318}{6271661}a^{8}+\frac{317862616}{6271661}a^{7}-\frac{267620157}{6271661}a^{6}+\frac{25861263}{6271661}a^{5}+\frac{197962304}{6271661}a^{4}-\frac{30086313}{570151}a^{3}+\frac{260496977}{6271661}a^{2}-\frac{118385594}{6271661}a+\frac{25902828}{6271661}$, $\frac{5047531}{6271661}a^{15}-\frac{3715427}{6271661}a^{14}+\frac{1570702}{6271661}a^{13}+\frac{17837151}{6271661}a^{12}-\frac{31900113}{6271661}a^{11}+\frac{87942721}{6271661}a^{10}-\frac{25326776}{6271661}a^{9}-\frac{58764048}{6271661}a^{8}+\frac{314391433}{6271661}a^{7}-\frac{526428502}{6271661}a^{6}+\frac{661719550}{6271661}a^{5}-\frac{611327582}{6271661}a^{4}+\frac{37893724}{570151}a^{3}-\frac{222352767}{6271661}a^{2}+\frac{67416534}{6271661}a-\frac{9106728}{6271661}$, $\frac{12225665}{6271661}a^{15}-\frac{16546132}{6271661}a^{14}-\frac{3825351}{6271661}a^{13}+\frac{52274262}{6271661}a^{12}-\frac{96886751}{6271661}a^{11}+\frac{207998784}{6271661}a^{10}-\frac{111398019}{6271661}a^{9}-\frac{297848019}{6271661}a^{8}+\frac{883233640}{6271661}a^{7}-\frac{1436946488}{6271661}a^{6}+\frac{1567324105}{6271661}a^{5}-\frac{1261829839}{6271661}a^{4}+\frac{64528415}{570151}a^{3}-\frac{270794911}{6271661}a^{2}+\frac{65082021}{6271661}a-\frac{2043176}{6271661}$, $\frac{883687}{6271661}a^{15}-\frac{12353601}{6271661}a^{14}+\frac{12702554}{6271661}a^{13}+\frac{9694600}{6271661}a^{12}-\frac{54292391}{6271661}a^{11}+\frac{95606277}{6271661}a^{10}-\frac{181870274}{6271661}a^{9}+\frac{42894178}{6271661}a^{8}+\frac{352389230}{6271661}a^{7}-\frac{864465886}{6271661}a^{6}+\frac{1283910772}{6271661}a^{5}-\frac{1274891003}{6271661}a^{4}+\frac{83854331}{570151}a^{3}-\frac{442545041}{6271661}a^{2}+\frac{121698398}{6271661}a-\frac{15871015}{6271661}$ | 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: | \( 65.9381053111 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 65.9381053111 \cdot 1}{2\cdot\sqrt{99038527051792384}}\cr\approx \mathstrut & 0.254473948593 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 16T42):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4\wr C_2$ |
Character table for $C_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-14}) \), 4.0.392.1 x2, 4.2.448.1 x2, \(\Q(\sqrt{2}, \sqrt{-7})\), 8.0.4917248.1, 8.0.314703872.3, 8.0.9834496.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 8 siblings: | 8.0.4917248.1, 8.0.314703872.3 |
Degree 16 sibling: | 16.4.2069703095939497984.1 |
Minimal sibling: | 8.0.4917248.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.22.5 | $x^{8} + 20 x^{7} + 154 x^{6} + 592 x^{5} + 1315 x^{4} + 1968 x^{3} + 2050 x^{2} + 828 x + 111$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ |
2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
\(7\) | 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |