Normalized defining polynomial
\( x^{15} - 3 x^{14} + 5 x^{13} - 2 x^{12} - 8 x^{11} + 18 x^{10} - 18 x^{9} + 20 x^{7} - 27 x^{6} + \cdots + 1 \)
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: | \(6114596477365737\) \(\medspace = 3^{4}\cdot 15601\cdot 4838718377\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.28\) | 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: | $3^{1/2}15601^{1/2}4838718377^{1/2}\approx 15048805.141895186$ | ||
Ramified primes: | \(3\), \(15601\), \(4838718377\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{75488845399577}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$
Monogenic: | Yes | |
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)
| |
Fundamental units: | $a^{14}-3a^{13}+4a^{12}-11a^{10}+18a^{9}-13a^{8}-8a^{7}+27a^{6}-24a^{5}+9a^{4}+11a^{3}-13a^{2}+9a-4$, $a$, $4a^{14}-9a^{13}+13a^{12}+2a^{11}-30a^{10}+47a^{9}-33a^{8}-26a^{7}+56a^{6}-56a^{5}+10a^{4}+20a^{3}-22a^{2}+16a-4$, $2a^{14}-5a^{13}+7a^{12}+a^{11}-18a^{10}+28a^{9}-19a^{8}-16a^{7}+38a^{6}-35a^{5}+9a^{4}+15a^{3}-16a^{2}+13a-5$, $3a^{14}-8a^{13}+12a^{12}-a^{11}-26a^{10}+46a^{9}-37a^{8}-16a^{7}+58a^{6}-62a^{5}+23a^{4}+19a^{3}-28a^{2}+22a-8$, $a^{13}-2a^{12}+2a^{11}+3a^{10}-10a^{9}+11a^{8}-2a^{7}-15a^{6}+19a^{5}-11a^{4}-5a^{3}+10a^{2}-8a+3$, $3a^{14}-7a^{13}+10a^{12}+2a^{11}-26a^{10}+41a^{9}-29a^{8}-23a^{7}+55a^{6}-56a^{5}+15a^{4}+21a^{3}-27a^{2}+21a-7$, $3a^{14}-7a^{13}+10a^{12}+2a^{11}-26a^{10}+41a^{9}-29a^{8}-23a^{7}+55a^{6}-56a^{5}+15a^{4}+21a^{3}-26a^{2}+21a-7$ | 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: | \( 53.4601866446 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 53.4601866446 \cdot 1}{2\cdot\sqrt{6114596477365737}}\cr\approx \mathstrut & 0.168261893271 \end{aligned}\]
Galois group
A non-solvable group of order 1307674368000 |
The 176 conjugacy class representatives for $S_{15}$ |
Character table for $S_{15}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 30 sibling: | 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 | $15$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $15$ | $15$ | $15$ | ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $15$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $15$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $15$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(15601\) | $\Q_{15601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{15601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(4838718377\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |