Normalized defining polynomial
\( x^{16} - 5 x^{15} + 12 x^{14} - 19 x^{13} + 21 x^{12} - 16 x^{11} + 6 x^{10} + 12 x^{9} - 28 x^{8} + \cdots + 1 \)
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: | \(3345005787800625\) \(\medspace = 3^{8}\cdot 5^{4}\cdot 13^{8}\) | 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: | \(9.34\) | 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: | $3^{1/2}5^{1/2}13^{1/2}\approx 13.96424004376894$ | ||
Ramified primes: | \(3\), \(5\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4201259}a^{15}-\frac{844160}{4201259}a^{14}+\frac{1138268}{4201259}a^{13}-\frac{476410}{4201259}a^{12}-\frac{1634204}{4201259}a^{11}+\frac{273623}{4201259}a^{10}+\frac{795002}{4201259}a^{9}-\frac{1897}{4201259}a^{8}+\frac{682328}{4201259}a^{7}+\frac{2016081}{4201259}a^{6}-\frac{1049503}{4201259}a^{5}-\frac{1487939}{4201259}a^{4}+\frac{743343}{4201259}a^{3}-\frac{867204}{4201259}a^{2}+\frac{2016916}{4201259}a-\frac{107421}{4201259}$
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: | \( \frac{8947}{8033} a^{15} - \frac{40788}{8033} a^{14} + \frac{87386}{8033} a^{13} - \frac{122437}{8033} a^{12} + \frac{114059}{8033} a^{11} - \frac{64231}{8033} a^{10} - \frac{1220}{8033} a^{9} + \frac{121765}{8033} a^{8} - \frac{194988}{8033} a^{7} + \frac{72428}{8033} a^{6} + \frac{87250}{8033} a^{5} - \frac{166072}{8033} a^{4} + \frac{161088}{8033} a^{3} - \frac{64577}{8033} a^{2} + \frac{24285}{8033} a - \frac{3468}{8033} \) (order $6$) | 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: | $\frac{796875}{4201259}a^{15}-\frac{1213956}{4201259}a^{14}-\frac{2908118}{4201259}a^{13}+\frac{11752044}{4201259}a^{12}-\frac{21469083}{4201259}a^{11}+\frac{23193579}{4201259}a^{10}-\frac{15833414}{4201259}a^{9}+\frac{9183883}{4201259}a^{8}+\frac{15788997}{4201259}a^{7}-\frac{43108574}{4201259}a^{6}+\frac{26127264}{4201259}a^{5}+\frac{7333168}{4201259}a^{4}-\frac{30267134}{4201259}a^{3}+\frac{32911705}{4201259}a^{2}-\frac{12108058}{4201259}a+\frac{3744009}{4201259}$, $\frac{1962229}{4201259}a^{15}-\frac{9047969}{4201259}a^{14}+\frac{18754684}{4201259}a^{13}-\frac{24384095}{4201259}a^{12}+\frac{20878732}{4201259}a^{11}-\frac{9914533}{4201259}a^{10}-\frac{1902350}{4201259}a^{9}+\frac{25174615}{4201259}a^{8}-\frac{40649152}{4201259}a^{7}+\frac{6299933}{4201259}a^{6}+\frac{28920728}{4201259}a^{5}-\frac{28919017}{4201259}a^{4}+\frac{24494445}{4201259}a^{3}-\frac{8502428}{4201259}a^{2}+\frac{2068879}{4201259}a-\frac{3236120}{4201259}$, $\frac{3231593}{4201259}a^{15}-\frac{11650482}{4201259}a^{14}+\frac{19187251}{4201259}a^{13}-\frac{20263098}{4201259}a^{12}+\frac{9588762}{4201259}a^{11}+\frac{3390968}{4201259}a^{10}-\frac{14200458}{4201259}a^{9}+\frac{41317550}{4201259}a^{8}-\frac{32800164}{4201259}a^{7}-\frac{20568620}{4201259}a^{6}+\frac{38421018}{4201259}a^{5}-\frac{34519996}{4201259}a^{4}+\frac{15573192}{4201259}a^{3}+\frac{12043755}{4201259}a^{2}-\frac{2591707}{4201259}a+\frac{4878258}{4201259}$, $\frac{4648396}{4201259}a^{15}-\frac{18262619}{4201259}a^{14}+\frac{33827233}{4201259}a^{13}-\frac{41914424}{4201259}a^{12}+\frac{32910217}{4201259}a^{11}-\frac{14701024}{4201259}a^{10}-\frac{6318493}{4201259}a^{9}+\frac{55051796}{4201259}a^{8}-\frac{64352529}{4201259}a^{7}-\frac{3934792}{4201259}a^{6}+\frac{42617461}{4201259}a^{5}-\frac{47419252}{4201259}a^{4}+\frac{43968314}{4201259}a^{3}-\frac{12198061}{4201259}a^{2}+\frac{8116329}{4201259}a-\frac{3110789}{4201259}$, $\frac{173680}{4201259}a^{15}-\frac{2373477}{4201259}a^{14}+\frac{8345254}{4201259}a^{13}-\frac{15897831}{4201259}a^{12}+\frac{21111097}{4201259}a^{11}-\frac{18604204}{4201259}a^{10}+\frac{9972843}{4201259}a^{9}+\frac{2428501}{4201259}a^{8}-\frac{23393127}{4201259}a^{7}+\frac{28425538}{4201259}a^{6}-\frac{6059325}{4201259}a^{5}-\frac{14206948}{4201259}a^{4}+\frac{28531983}{4201259}a^{3}-\frac{26063124}{4201259}a^{2}+\frac{5397978}{4201259}a-\frac{3289320}{4201259}$, $\frac{1440473}{4201259}a^{15}-\frac{6691533}{4201259}a^{14}+\frac{14769575}{4201259}a^{13}-\frac{22096870}{4201259}a^{12}+\frac{22704388}{4201259}a^{11}-\frac{15575701}{4201259}a^{10}+\frac{3938985}{4201259}a^{9}+\frac{19247364}{4201259}a^{8}-\frac{34689460}{4201259}a^{7}+\frac{19371376}{4201259}a^{6}+\frac{4504900}{4201259}a^{5}-\frac{25864966}{4201259}a^{4}+\frac{32652499}{4201259}a^{3}-\frac{19407763}{4201259}a^{2}+\frac{12203739}{4201259}a-\frac{479904}{4201259}$, $\frac{1130760}{4201259}a^{15}-\frac{3713023}{4201259}a^{14}+\frac{6015181}{4201259}a^{13}-\frac{7338843}{4201259}a^{12}+\frac{6048556}{4201259}a^{11}-\frac{3976834}{4201259}a^{10}+\frac{469513}{4201259}a^{9}+\frac{10194147}{4201259}a^{8}-\frac{7804811}{4201259}a^{7}-\frac{4413315}{4201259}a^{6}+\frac{2019968}{4201259}a^{5}-\frac{6906874}{4201259}a^{4}+\frac{13447586}{4201259}a^{3}-\frac{4738145}{4201259}a^{2}+\frac{7091787}{4201259}a-\frac{569752}{4201259}$ | 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: | \( 22.9306095521 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22.9306095521 \cdot 1}{6\cdot\sqrt{3345005787800625}}\cr\approx \mathstrut & 0.160511021714 \end{aligned}\]
Galois group
A solvable group of order 32 |
The 14 conjugacy class representatives for $D_8:C_2$ |
Character table for $D_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), 4.0.117.1 x2, 4.2.507.1 x2, \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.2313441.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 siblings: | 16.4.232292068597265625.1, 16.0.12370583534765625.1 |
Minimal sibling: | 16.0.12370583534765625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.8.0.1 | $x^{8} + x^{4} + 3 x^{2} + 4 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
5.8.4.2 | $x^{8} + 100 x^{4} - 500 x^{2} + 1250$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
\(13\) | 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |