Normalized defining polynomial
\( x^{15} - 4 x^{14} + 5 x^{13} + x^{12} - 7 x^{11} + x^{10} + 7 x^{9} - 2 x^{8} - 8 x^{7} + 22 x^{6} + \cdots + 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(11354350039657729\) \(\medspace = 13^{2}\cdot 127^{2}\cdot 1609^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.76\) | 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: | $13^{2/3}127^{2/3}1609^{1/2}\approx 5603.285322545042$ | ||
Ramified primes: | \(13\), \(127\), \(1609\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1609}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{679097}a^{14}-\frac{18368}{679097}a^{13}-\frac{201252}{679097}a^{12}+\frac{145855}{679097}a^{11}-\frac{122659}{679097}a^{10}-\frac{54872}{679097}a^{9}-\frac{110533}{679097}a^{8}+\frac{7077}{679097}a^{7}-\frac{254509}{679097}a^{6}+\frac{257744}{679097}a^{5}+\frac{95237}{679097}a^{4}-\frac{257458}{679097}a^{3}+\frac{85378}{679097}a^{2}+\frac{153392}{679097}a+\frac{3661}{679097}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{174629}{679097}a^{14}-\frac{210341}{679097}a^{13}-\frac{486661}{679097}a^{12}+\frac{979810}{679097}a^{11}+\frac{259063}{679097}a^{10}-\frac{1542012}{679097}a^{9}-\frac{293226}{679097}a^{8}+\frac{1251087}{679097}a^{7}-\frac{469899}{679097}a^{6}+\frac{1065107}{679097}a^{5}+\frac{735640}{679097}a^{4}-\frac{3411682}{679097}a^{3}+\frac{2616515}{679097}a^{2}-\frac{289597}{679097}a+\frac{965589}{679097}$, $\frac{230384}{679097}a^{14}-\frac{239905}{679097}a^{13}-\frac{572190}{679097}a^{12}+\frac{938760}{679097}a^{11}+\frac{592405}{679097}a^{10}-\frac{1598387}{679097}a^{9}-\frac{934463}{679097}a^{8}+\frac{1273865}{679097}a^{7}-\frac{208282}{679097}a^{6}+\frac{1210210}{679097}a^{5}+\frac{815132}{679097}a^{4}-\frac{2550989}{679097}a^{3}+\frac{1038741}{679097}a^{2}+\frac{212842}{679097}a+\frac{676447}{679097}$, $\frac{39462}{679097}a^{14}+\frac{437580}{679097}a^{13}-\frac{1125203}{679097}a^{12}+\frac{382935}{679097}a^{11}+\frac{1592152}{679097}a^{10}-\frac{1076725}{679097}a^{9}-\frac{2050506}{679097}a^{8}+\frac{842804}{679097}a^{7}+\frac{1089569}{679097}a^{6}-\frac{2458429}{679097}a^{5}+\frac{5552472}{679097}a^{4}-\frac{5949252}{679097}a^{3}+\frac{2902807}{679097}a^{2}-\frac{994651}{679097}a+\frac{1860012}{679097}$, $\frac{91140}{679097}a^{14}-\frac{85415}{679097}a^{13}-\frac{376407}{679097}a^{12}+\frac{580022}{679097}a^{11}+\frac{153554}{679097}a^{10}-\frac{842869}{679097}a^{9}-\frac{252722}{679097}a^{8}+\frac{534727}{679097}a^{7}-\frac{34031}{679097}a^{6}+\frac{822930}{679097}a^{5}+\frac{1040520}{679097}a^{4}-\frac{2599867}{679097}a^{3}+\frac{2294785}{679097}a^{2}-\frac{1102156}{679097}a+\frac{906010}{679097}$, $\frac{140071}{679097}a^{14}-\frac{404692}{679097}a^{13}+\frac{426675}{679097}a^{12}+\frac{101557}{679097}a^{11}-\frac{493786}{679097}a^{10}+\frac{43934}{679097}a^{9}+\frac{264660}{679097}a^{8}-\frac{199153}{679097}a^{7}-\frac{812221}{679097}a^{6}+\frac{2342401}{679097}a^{5}-\frac{3635126}{679097}a^{4}+\frac{3763055}{679097}a^{3}-\frac{2632720}{679097}a^{2}+\frac{1858140}{679097}a-\frac{597401}{679097}$, $\frac{488175}{679097}a^{14}-\frac{1359806}{679097}a^{13}+\frac{805181}{679097}a^{12}+\frac{1481466}{679097}a^{11}-\frac{1716641}{679097}a^{10}-\frac{1515629}{679097}a^{9}+\frac{1600345}{679097}a^{8}+\frac{927133}{679097}a^{7}-\frac{2776731}{679097}a^{6}+\frac{7196913}{679097}a^{5}-\frac{9523697}{679097}a^{4}+\frac{6109095}{679097}a^{3}-\frac{2210516}{679097}a^{2}+\frac{2187992}{679097}a-\frac{174629}{679097}$, $\frac{152986}{679097}a^{14}-\frac{622559}{679097}a^{13}+\frac{840411}{679097}a^{12}+\frac{3804}{679097}a^{11}-\frac{980567}{679097}a^{10}+\frac{349322}{679097}a^{9}+\frac{871956}{679097}a^{8}-\frac{477793}{679097}a^{7}-\frac{966476}{679097}a^{6}+\frac{3530861}{679097}a^{5}-\frac{6210326}{679097}a^{4}+\frac{6268285}{679097}a^{3}-\frac{4187572}{679097}a^{2}+\frac{1989871}{679097}a-\frac{1531473}{679097}$, $\frac{238798}{679097}a^{14}-\frac{633238}{679097}a^{13}+\frac{440497}{679097}a^{12}+\frac{355354}{679097}a^{11}-\frac{591175}{679097}a^{10}-\frac{147241}{679097}a^{9}+\frac{82862}{679097}a^{8}-\frac{298987}{679097}a^{7}-\frac{454167}{679097}a^{6}+\frac{4227893}{679097}a^{5}-\frac{5307083}{679097}a^{4}+\frac{4307799}{679097}a^{3}-\frac{2470878}{679097}a^{2}+\frac{1247927}{679097}a-\frac{437458}{679097}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 79.4890458165 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 79.4890458165 \cdot 1}{2\cdot\sqrt{11354350039657729}}\cr\approx \mathstrut & 0.183596900795 \end{aligned}\]
Galois group
$C_3\wr S_5$ (as 15T78):
A non-solvable group of order 29160 |
The 108 conjugacy class representatives for $C_3\wr S_5$ |
Character table for $C_3\wr S_5$ |
Intermediate fields
5.1.1609.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 15 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 45 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $15$ | $15$ | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.3.0.1}{3} }^{5}$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(127\) | 127.3.2.2 | $x^{3} + 1143$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
127.3.0.1 | $x^{3} + 3 x + 124$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
127.3.0.1 | $x^{3} + 3 x + 124$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
127.6.0.1 | $x^{6} + 84 x^{3} + 115 x^{2} + 82 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(1609\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |