Normalized defining polynomial
\( x^{14} - x^{13} - 5 x^{12} - 2 x^{11} + 9 x^{10} + 13 x^{9} + x^{8} - 7 x^{7} + x^{6} + 13 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(922560566965027225\) \(\medspace = 5^{2}\cdot 577^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.20\) | 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: | $5^{1/2}577^{1/2}\approx 53.71219600798314$ | ||
Ramified primes: | \(5\), \(577\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{1}{5}a^{11}+\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $a$, $\frac{4}{5}a^{13}-\frac{6}{5}a^{12}-\frac{11}{5}a^{11}-\frac{9}{5}a^{10}+\frac{9}{5}a^{9}+\frac{44}{5}a^{8}+\frac{38}{5}a^{7}+\frac{19}{5}a^{6}-\frac{11}{5}a^{5}+\frac{16}{5}a^{4}+\frac{44}{5}a^{3}+\frac{34}{5}a^{2}+\frac{4}{5}a-3$, $\frac{2}{5}a^{13}-\frac{4}{5}a^{12}-\frac{9}{5}a^{11}+\frac{7}{5}a^{10}+\frac{28}{5}a^{9}+\frac{6}{5}a^{8}-\frac{32}{5}a^{7}-4a^{6}+\frac{16}{5}a^{5}+\frac{17}{5}a^{4}-\frac{17}{5}a^{3}-\frac{24}{5}a^{2}-\frac{9}{5}a+\frac{4}{5}$, $\frac{9}{5}a^{13}-\frac{11}{5}a^{12}-\frac{41}{5}a^{11}-\frac{9}{5}a^{10}+\frac{74}{5}a^{9}+\frac{89}{5}a^{8}+\frac{3}{5}a^{7}-\frac{31}{5}a^{6}+\frac{29}{5}a^{5}+\frac{101}{5}a^{4}+\frac{64}{5}a^{3}-\frac{1}{5}a^{2}-\frac{21}{5}a-1$, $\frac{13}{5}a^{13}-3a^{12}-13a^{11}-\frac{6}{5}a^{10}+\frac{116}{5}a^{9}+25a^{8}-\frac{12}{5}a^{7}-\frac{58}{5}a^{6}+12a^{5}+\frac{139}{5}a^{4}+\frac{76}{5}a^{3}-4a^{2}-5a+\frac{2}{5}$, $\frac{7}{5}a^{13}-a^{12}-8a^{11}-\frac{14}{5}a^{10}+\frac{59}{5}a^{9}+19a^{8}+\frac{22}{5}a^{7}-\frac{42}{5}a^{6}+4a^{5}+\frac{86}{5}a^{4}+\frac{74}{5}a^{3}-a^{2}-4a-\frac{7}{5}$, $\frac{7}{5}a^{13}-\frac{7}{5}a^{12}-\frac{32}{5}a^{11}-\frac{16}{5}a^{10}+\frac{46}{5}a^{9}+\frac{83}{5}a^{8}+7a^{7}-\frac{11}{5}a^{6}+\frac{13}{5}a^{5}+\frac{84}{5}a^{4}+\frac{81}{5}a^{3}+\frac{23}{5}a^{2}-\frac{17}{5}a-\frac{14}{5}$, $\frac{13}{5}a^{13}-\frac{13}{5}a^{12}-\frac{63}{5}a^{11}-\frac{24}{5}a^{10}+\frac{99}{5}a^{9}+\frac{152}{5}a^{8}+8a^{7}-\frac{39}{5}a^{6}+\frac{32}{5}a^{5}+\frac{141}{5}a^{4}+\frac{114}{5}a^{3}+\frac{22}{5}a^{2}-\frac{18}{5}a-\frac{11}{5}$, $2a^{13}-\frac{18}{5}a^{12}-\frac{33}{5}a^{11}+\frac{2}{5}a^{10}+\frac{78}{5}a^{9}+\frac{72}{5}a^{8}-\frac{28}{5}a^{7}-\frac{36}{5}a^{6}+\frac{32}{5}a^{5}+\frac{107}{5}a^{4}+\frac{18}{5}a^{3}-\frac{23}{5}a^{2}-\frac{28}{5}a+\frac{12}{5}$ | 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: | \( 5423.89330462 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{4}\cdot 5423.89330462 \cdot 1}{2\cdot\sqrt{922560566965027225}}\cr\approx \mathstrut & 0.281632743430 \end{aligned}\]
Galois group
$C_2^6:D_7$ (as 14T28):
A solvable group of order 896 |
The 20 conjugacy class representatives for $C_2^6:D_7$ |
Character table for $C_2^6:D_7$ |
Intermediate fields
7.7.192100033.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 28 siblings: | data not computed |
Minimal sibling: | 14.6.532317447138820708825.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{5}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.7.0.1}{7} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(577\) | $\Q_{577}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{577}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |