Normalized defining polynomial
\( x^{15} - 14x^{10} + 35x^{5} + 7 \)
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)
| |
Discriminant: | \(1324654439158203125\) \(\medspace = 5^{9}\cdot 7^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.15\) | 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: | $5^{3/4}7^{14/15}\approx 20.558227794588174$ | ||
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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41}a^{10}+\frac{19}{41}a^{5}+\frac{6}{41}$, $\frac{1}{41}a^{11}+\frac{19}{41}a^{6}+\frac{6}{41}a$, $\frac{1}{205}a^{12}-\frac{1}{205}a^{11}+\frac{1}{205}a^{10}+\frac{2}{5}a^{9}+\frac{19}{205}a^{7}+\frac{22}{205}a^{6}-\frac{22}{205}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{47}{205}a^{2}+\frac{7}{41}a+\frac{6}{205}$, $\frac{1}{205}a^{13}-\frac{2}{205}a^{10}+\frac{2}{5}a^{9}+\frac{19}{205}a^{8}+\frac{1}{5}a^{7}-\frac{38}{205}a^{5}+\frac{1}{5}a^{4}-\frac{76}{205}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{94}{205}$, $\frac{1}{205}a^{14}-\frac{2}{205}a^{11}+\frac{2}{205}a^{10}+\frac{19}{205}a^{9}+\frac{1}{5}a^{8}-\frac{38}{205}a^{6}-\frac{44}{205}a^{5}-\frac{76}{205}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{94}{205}a-\frac{14}{41}$
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)
| |
Fundamental units: | $\frac{3}{41}a^{10}-\frac{25}{41}a^{5}-\frac{23}{41}$, $\frac{3}{41}a^{10}-\frac{25}{41}a^{5}+\frac{18}{41}$, $\frac{4}{205}a^{14}-\frac{1}{41}a^{13}+\frac{6}{205}a^{12}+\frac{6}{205}a^{11}-\frac{1}{205}a^{10}-\frac{47}{205}a^{9}+\frac{69}{205}a^{8}-\frac{91}{205}a^{7}-\frac{10}{41}a^{6}+\frac{22}{205}a^{5}+\frac{13}{41}a^{4}-\frac{47}{41}a^{3}+\frac{241}{205}a^{2}-\frac{46}{205}a+\frac{76}{205}$, $\frac{4}{205}a^{14}-\frac{1}{41}a^{13}+\frac{6}{205}a^{12}+\frac{6}{205}a^{11}-\frac{1}{205}a^{10}-\frac{47}{205}a^{9}+\frac{69}{205}a^{8}-\frac{91}{205}a^{7}-\frac{10}{41}a^{6}+\frac{22}{205}a^{5}+\frac{13}{41}a^{4}-\frac{47}{41}a^{3}+\frac{241}{205}a^{2}-\frac{46}{205}a-\frac{129}{205}$, $\frac{2}{41}a^{14}-\frac{13}{205}a^{13}+\frac{4}{205}a^{12}+\frac{1}{205}a^{11}+\frac{1}{41}a^{10}-\frac{138}{205}a^{9}+\frac{163}{205}a^{8}-\frac{47}{205}a^{7}-\frac{22}{205}a^{6}-\frac{69}{205}a^{5}+\frac{388}{205}a^{4}-\frac{324}{205}a^{3}+\frac{147}{205}a^{2}+\frac{47}{205}a+\frac{71}{205}$, $\frac{2}{205}a^{14}-\frac{8}{205}a^{13}-\frac{3}{205}a^{12}+\frac{4}{205}a^{11}+\frac{2}{205}a^{10}-\frac{44}{205}a^{9}+\frac{27}{41}a^{8}+\frac{5}{41}a^{7}-\frac{47}{205}a^{6}-\frac{3}{205}a^{5}+\frac{258}{205}a^{4}-\frac{499}{205}a^{3}+\frac{21}{41}a^{2}+\frac{24}{205}a-\frac{111}{205}$, $\frac{4}{205}a^{14}-\frac{1}{41}a^{13}+\frac{6}{205}a^{12}+\frac{6}{205}a^{11}+\frac{4}{205}a^{10}-\frac{47}{205}a^{9}+\frac{69}{205}a^{8}-\frac{91}{205}a^{7}-\frac{10}{41}a^{6}-\frac{88}{205}a^{5}+\frac{13}{41}a^{4}-\frac{47}{41}a^{3}+\frac{241}{205}a^{2}-\frac{46}{205}a+\frac{106}{205}$, $\frac{1}{41}a^{14}-\frac{12}{205}a^{13}+\frac{3}{41}a^{12}-\frac{1}{205}a^{10}-\frac{69}{205}a^{9}+\frac{182}{205}a^{8}-\frac{207}{205}a^{7}-\frac{19}{205}a^{5}+\frac{153}{205}a^{4}-\frac{523}{205}a^{3}+\frac{541}{205}a^{2}-\frac{2}{5}a-\frac{47}{205}$ | 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: | \( 1272.85837366 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 1272.85837366 \cdot 1}{2\cdot\sqrt{1324654439158203125}}\cr\approx \mathstrut & 0.272187380385 \end{aligned}\]
Galois group
$C_3\times F_5$ (as 15T8):
A solvable group of order 60 |
The 15 conjugacy class representatives for $F_5\times C_3$ |
Character table for $F_5\times C_3$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 5.1.300125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | R | $15$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | $15$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
\(7\) | 7.15.14.1 | $x^{15} + 7$ | $15$ | $1$ | $14$ | $F_5\times C_3$ | $[\ ]_{15}^{4}$ |