Normalized defining polynomial
\( x^{14} - x^{13} + 6 x^{12} - 4 x^{11} + 12 x^{10} - 2 x^{9} + 11 x^{8} + 3 x^{7} + 11 x^{6} - 2 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-445660887761159\) \(\medspace = -\,7^{2}\cdot 71^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(11.13\) | 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: | $7^{1/2}71^{3/4}\approx 64.71318596135296$ | ||
Ramified primes: | \(7\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-71}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{1}{7}a^{11}+\frac{2}{7}a^{9}+\frac{2}{7}a^{8}+\frac{2}{7}a^{6}+\frac{2}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{13}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}-\frac{2}{7}a^{8}+\frac{2}{7}a^{7}-\frac{2}{7}a^{6}+\frac{2}{7}a^{5}-\frac{2}{7}a^{3}+\frac{1}{7}a^{2}-\frac{1}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{5}{7}a^{13}+\frac{12}{7}a^{12}+2a^{11}+\frac{73}{7}a^{10}+\frac{3}{7}a^{9}+21a^{8}+\frac{45}{7}a^{7}+18a^{6}+\frac{94}{7}a^{5}+\frac{115}{7}a^{4}-a^{3}+\frac{103}{7}a^{2}-\frac{23}{7}a+4$, $\frac{12}{7}a^{13}-\frac{9}{7}a^{12}+9a^{11}-\frac{25}{7}a^{10}+\frac{101}{7}a^{9}+4a^{8}+\frac{80}{7}a^{7}+9a^{6}+\frac{115}{7}a^{5}-\frac{11}{7}a^{4}+14a^{3}+\frac{12}{7}a^{2}+\frac{19}{7}a+2$, $\frac{4}{7}a^{13}-\frac{10}{7}a^{12}+4a^{11}-\frac{48}{7}a^{10}+\frac{57}{7}a^{9}-9a^{8}+\frac{22}{7}a^{7}-6a^{6}+\frac{1}{7}a^{5}-\frac{83}{7}a^{4}+5a^{3}-\frac{73}{7}a^{2}+\frac{11}{7}a-3$, $a$, $\frac{1}{7}a^{13}-\frac{3}{7}a^{12}+\frac{10}{7}a^{11}-\frac{19}{7}a^{10}+\frac{29}{7}a^{9}-\frac{36}{7}a^{8}+\frac{23}{7}a^{7}-\frac{22}{7}a^{6}+\frac{2}{7}a^{5}-\frac{27}{7}a^{4}+\frac{27}{7}a^{3}-\frac{41}{7}a^{2}+\frac{25}{7}a-\frac{11}{7}$, $\frac{4}{7}a^{13}+\frac{1}{7}a^{12}+\frac{18}{7}a^{11}+\frac{15}{7}a^{10}+\frac{23}{7}a^{9}+\frac{57}{7}a^{8}+\frac{29}{7}a^{7}+\frac{71}{7}a^{6}+\frac{64}{7}a^{5}+\frac{44}{7}a^{4}+\frac{43}{7}a^{3}+\frac{53}{7}a^{2}+\frac{1}{7}a+\frac{25}{7}$ | 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: | \( 25.684570553 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 25.684570553 \cdot 1}{2\cdot\sqrt{445660887761159}}\cr\approx \mathstrut & 0.23517932038 \end{aligned}\]
Galois group
$C_2^6:D_7$ (as 14T27):
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
7.1.357911.1 |
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: | 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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(71\) | 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
71.4.3.1 | $x^{4} + 71$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |