Normalized defining polynomial
\( x^{15} - x^{10} - 2x^{5} + 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(8620460479736328125\) \(\medspace = 5^{15}\cdot 7^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.30\) | 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: | $5^{123/100}7^{2/3}\approx 26.493040212237457$ | ||
Ramified primes: | \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
| |
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: | $a$, $a^{13}-a^{8}-a^{3}$, $a^{13}-a^{12}-2a^{3}+2a^{2}$, $a^{14}+a^{13}-a^{9}-a^{8}-a^{4}-a^{3}$, $a^{12}+a^{10}-a^{7}-a^{5}-a^{2}+a-1$, $a^{11}-a^{6}+a^{2}-2a+1$, $a^{11}-a^{2}-2a-1$, $a^{13}+a^{11}-a^{7}-2a^{3}-a-1$ | 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: | \( 3529.85179224 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 3529.85179224 \cdot 1}{2\cdot\sqrt{8620460479736328125}}\cr\approx \mathstrut & 0.295890103632 \end{aligned}\]
Galois group
$C_5^2:C_{12}$ (as 15T19):
A solvable group of order 300 |
The 14 conjugacy class representatives for $(C_5^2 : C_4):C_3$ |
Character table for $(C_5^2 : C_4):C_3$ |
Intermediate fields
\(\Q(\zeta_{7})^+\) |
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 25 sibling: | data not computed |
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{5}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.15.15.39 | $x^{15} - 60 x^{12} + 15 x^{10} + 900 x^{9} - 300 x^{8} - 675 x^{7} + 9000 x^{6} + 22575 x^{5} + 14250 x^{4} - 125 x^{3} - 1875 x^{2} + 125$ | $5$ | $3$ | $15$ | $(C_5^2 : C_4):C_3$ | $[5/4, 5/4]_{4}^{3}$ |
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |