Normalized defining polynomial
\( x^{15} - 20 x^{13} - 20 x^{12} + 160 x^{11} + 320 x^{10} - 500 x^{9} - 1920 x^{8} - 400 x^{7} + \cdots - 16 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-395638255629389500000000000000\) \(\medspace = -\,2^{14}\cdot 5^{15}\cdot 7^{6}\cdot 13\cdot 517364567\) | 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: | \(94.01\) | 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))
| |
Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(5\), \(7\), \(13\), \(517364567\) | 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: | $\Q(\sqrt{-33628696855}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{1}{8}a^{12}-2a^{10}-\frac{7}{4}a^{9}+12a^{8}+21a^{7}-\frac{49}{2}a^{6}-84a^{5}-28a^{4}+101a^{3}+120a^{2}+44a+3$, $\frac{1}{2}a^{6}-4a^{4}-5a^{3}+8a^{2}+20a+11$, $\frac{1}{4}a^{12}-4a^{10}-\frac{17}{4}a^{9}+24a^{8}+51a^{7}-42a^{6}-204a^{5}-112a^{4}+240a^{3}+352a^{2}+128a+3$, $\frac{1}{4}a^{8}-3a^{6}-\frac{5}{2}a^{5}+12a^{4}+20a^{3}-11a^{2}-40a-19$, $\frac{1}{4}a^{9}-3a^{7}-4a^{6}+12a^{5}+32a^{4}+3a^{3}-64a^{2}-76a-23$, $\frac{1}{8}a^{14}+\frac{1}{4}a^{13}-\frac{7}{4}a^{12}-\frac{23}{4}a^{11}+\frac{19}{4}a^{10}+\frac{167}{4}a^{9}+\frac{153}{4}a^{8}-97a^{7}-\frac{467}{2}a^{6}-\frac{169}{2}a^{5}+275a^{4}+416a^{3}+250a^{2}+73a+11$, $\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{9}{4}a^{12}+\frac{7}{4}a^{11}+\frac{81}{4}a^{10}+\frac{29}{4}a^{9}-\frac{377}{4}a^{8}-118a^{7}+\frac{351}{2}a^{6}+\frac{893}{2}a^{5}+87a^{4}-520a^{3}-518a^{2}-137a+9$, $\frac{1}{2}a^{14}+\frac{9}{8}a^{13}-\frac{15}{2}a^{12}-\frac{107}{4}a^{11}+\frac{81}{4}a^{10}+204a^{9}+\frac{411}{2}a^{8}-490a^{7}-\frac{2569}{2}a^{6}-\frac{1093}{2}a^{5}+\frac{3133}{2}a^{4}+2573a^{3}+1583a^{2}+380a+13$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{15}{8}a^{12}+\frac{1}{4}a^{11}+11a^{10}+\frac{29}{4}a^{9}-\frac{49}{2}a^{8}-\frac{63}{2}a^{7}-4a^{6}-\frac{19}{2}a^{5}+21a^{4}+131a^{3}+139a^{2}+41a-1$, $\frac{13}{8}a^{14}-\frac{21}{8}a^{13}-\frac{113}{4}a^{12}+13a^{11}+\frac{955}{4}a^{10}+\frac{545}{4}a^{9}-1028a^{8}-1468a^{7}+\frac{3371}{2}a^{6}+\frac{9883}{2}a^{5}+1384a^{4}-5218a^{3}-5541a^{2}-1699a-45$, $\frac{1}{2}a^{14}-\frac{5}{4}a^{13}-\frac{75}{8}a^{12}+\frac{39}{4}a^{11}+\frac{353}{4}a^{10}+\frac{45}{2}a^{9}-\frac{1665}{4}a^{8}-\frac{987}{2}a^{7}+766a^{6}+\frac{3725}{2}a^{5}+334a^{4}-2082a^{3}-1993a^{2}-518a+17$ (assuming GRH) | 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: | \( 5019061959.44 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{3}\cdot 5019061959.44 \cdot 1}{2\cdot\sqrt{395638255629389500000000000000}}\cr\approx \mathstrut & 0.506702431340 \end{aligned}\] (assuming GRH)
Galois group
$S_3\wr F_5$ (as 15T87):
A solvable group of order 155520 |
The 63 conjugacy class representatives for $S_3\wr F_5$ |
Character table for $S_3\wr F_5$ |
Intermediate fields
5.5.2450000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 30 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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | R | $15$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.15.14.1 | $x^{15} + 2$ | $15$ | $1$ | $14$ | $C_{15} : C_4$ | $[\ ]_{15}^{4}$ |
\(5\) | 5.15.15.18 | $x^{15} - 60 x^{12} + 60 x^{11} + 15 x^{10} + 4425 x^{9} - 2400 x^{8} + 600 x^{7} + 17725 x^{6} + 88575 x^{5} - 1875 x^{4} - 4000 x^{3} + 4500 x^{2} + 1500 x + 125$ | $5$ | $3$ | $15$ | $F_5\times C_3$ | $[5/4]_{4}^{3}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.12.6.2 | $x^{12} + 49 x^{8} - 1715 x^{6} + 9604 x^{4} - 100842 x^{2} + 352947$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.12.0.1 | $x^{12} + x^{8} + 5 x^{7} + 8 x^{6} + 11 x^{5} + 3 x^{4} + x^{3} + x^{2} + 4 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(517364567\) | $\Q_{517364567}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |