Normalized defining polynomial
\( x^{14} - 3 x^{13} + 2 x^{12} + 4 x^{11} - 8 x^{10} + 2 x^{9} + 8 x^{8} - 10 x^{7} + x^{6} + 7 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(63097400703125\) \(\medspace = 5^{7}\cdot 807646729\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.68\) | 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: | $5^{1/2}807646729^{1/2}\approx 63547.09784876096$ | ||
Ramified primes: | \(5\), \(807646729\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{4038233645}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17}a^{13}+\frac{5}{17}a^{12}+\frac{8}{17}a^{11}-\frac{8}{17}a^{9}+\frac{6}{17}a^{8}+\frac{5}{17}a^{7}-\frac{4}{17}a^{6}+\frac{3}{17}a^{5}-\frac{3}{17}a^{4}+\frac{4}{17}a^{3}-\frac{1}{17}a^{2}-\frac{6}{17}a+\frac{2}{17}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $a$, $\frac{2}{17}a^{13}-\frac{7}{17}a^{12}-\frac{1}{17}a^{11}+2a^{10}-\frac{50}{17}a^{9}-\frac{5}{17}a^{8}+\frac{78}{17}a^{7}-\frac{59}{17}a^{6}-\frac{28}{17}a^{5}+\frac{79}{17}a^{4}-\frac{43}{17}a^{3}-\frac{19}{17}a^{2}+\frac{39}{17}a-\frac{13}{17}$, $\frac{5}{17}a^{13}-\frac{9}{17}a^{12}-\frac{11}{17}a^{11}+2a^{10}-\frac{6}{17}a^{9}-\frac{55}{17}a^{8}+\frac{59}{17}a^{7}+\frac{14}{17}a^{6}-\frac{70}{17}a^{5}+\frac{36}{17}a^{4}+\frac{37}{17}a^{3}-\frac{39}{17}a^{2}+\frac{4}{17}a+\frac{10}{17}$, $\frac{1}{17}a^{13}+\frac{5}{17}a^{12}-\frac{26}{17}a^{11}+2a^{10}+\frac{9}{17}a^{9}-\frac{62}{17}a^{8}+\frac{39}{17}a^{7}+\frac{47}{17}a^{6}-\frac{82}{17}a^{5}+\frac{14}{17}a^{4}+\frac{38}{17}a^{3}-\frac{18}{17}a^{2}-\frac{6}{17}a+\frac{2}{17}$, $\frac{15}{17}a^{13}-\frac{44}{17}a^{12}+\frac{35}{17}a^{11}+2a^{10}-\frac{86}{17}a^{9}+\frac{39}{17}a^{8}+\frac{58}{17}a^{7}-\frac{111}{17}a^{6}+\frac{45}{17}a^{5}+\frac{40}{17}a^{4}-\frac{59}{17}a^{3}+\frac{19}{17}a^{2}+\frac{12}{17}a-\frac{4}{17}$, $\frac{1}{17}a^{13}+\frac{5}{17}a^{12}-\frac{9}{17}a^{11}-a^{10}+\frac{43}{17}a^{9}-\frac{11}{17}a^{8}-\frac{46}{17}a^{7}+\frac{47}{17}a^{6}+\frac{3}{17}a^{5}-\frac{54}{17}a^{4}+\frac{38}{17}a^{3}-\frac{1}{17}a^{2}-\frac{23}{17}a+\frac{19}{17}$, $\frac{18}{17}a^{13}-\frac{46}{17}a^{12}+\frac{25}{17}a^{11}+3a^{10}-\frac{93}{17}a^{9}+\frac{23}{17}a^{8}+\frac{90}{17}a^{7}-\frac{123}{17}a^{6}+\frac{20}{17}a^{5}+\frac{65}{17}a^{4}-\frac{64}{17}a^{3}+\frac{16}{17}a^{2}+\frac{28}{17}a+\frac{2}{17}$ | 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: | \( 10.6889300961 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 10.6889300961 \cdot 1}{2\cdot\sqrt{63097400703125}}\cr\approx \mathstrut & 0.165591373434 \end{aligned}\]
Galois group
$S_7\wr C_2$ (as 14T61):
A non-solvable group of order 50803200 |
The 135 conjugacy class representatives for $S_7\wr C_2$ are not computed |
Character table for $S_7\wr C_2$ is not computed |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 28 siblings: | data not computed |
Degree 42 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
\(807646729\) | $\Q_{807646729}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{807646729}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |