Normalized defining polynomial
\( x^{14} - 3 x^{12} - 3 x^{11} + 6 x^{10} + 7 x^{9} - 5 x^{8} - 12 x^{7} + 2 x^{6} + 11 x^{5} + x^{4} + \cdots - 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(103886084489093\) \(\medspace = 53\cdot 977^{2}\cdot 1433^{2}\) | 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: | \(10.03\) | 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: | $53^{1/2}977^{1/2}1433^{1/2}\approx 8614.068318744634$ | ||
Ramified primes: | \(53\), \(977\), \(1433\) | 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: | \(\Q(\sqrt{53}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}-\frac{2}{7}a^{12}+\frac{1}{7}a^{11}+\frac{2}{7}a^{10}+\frac{2}{7}a^{9}+\frac{3}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{3}{7}a^{5}-\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}-\frac{3}{7}a+\frac{3}{7}$
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)
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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)
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Fundamental units: | $a$, $\frac{10}{7}a^{13}+\frac{8}{7}a^{12}-\frac{25}{7}a^{11}-\frac{50}{7}a^{10}+\frac{27}{7}a^{9}+\frac{93}{7}a^{8}+\frac{9}{7}a^{7}-\frac{124}{7}a^{6}-\frac{54}{7}a^{5}+\frac{85}{7}a^{4}+\frac{64}{7}a^{3}-\frac{44}{7}a^{2}-\frac{44}{7}a+\frac{16}{7}$, $\frac{10}{7}a^{13}+\frac{1}{7}a^{12}-\frac{32}{7}a^{11}-\frac{36}{7}a^{10}+\frac{62}{7}a^{9}+\frac{93}{7}a^{8}-\frac{40}{7}a^{7}-\frac{152}{7}a^{6}-\frac{19}{7}a^{5}+\frac{127}{7}a^{4}+\frac{57}{7}a^{3}-\frac{65}{7}a^{2}-\frac{51}{7}a+\frac{23}{7}$, $\frac{2}{7}a^{13}+\frac{3}{7}a^{12}-\frac{5}{7}a^{11}-\frac{17}{7}a^{10}-\frac{3}{7}a^{9}+\frac{34}{7}a^{8}+\frac{27}{7}a^{7}-\frac{29}{7}a^{6}-\frac{50}{7}a^{5}+\frac{3}{7}a^{4}+\frac{38}{7}a^{3}+\frac{15}{7}a^{2}-\frac{13}{7}a-\frac{8}{7}$, $\frac{19}{7}a^{13}+\frac{11}{7}a^{12}-\frac{51}{7}a^{11}-\frac{88}{7}a^{10}+\frac{66}{7}a^{9}+\frac{176}{7}a^{8}+\frac{1}{7}a^{7}-\frac{237}{7}a^{6}-\frac{90}{7}a^{5}+\frac{172}{7}a^{4}+\frac{109}{7}a^{3}-\frac{85}{7}a^{2}-\frac{85}{7}a+\frac{43}{7}$, $\frac{22}{7}a^{13}+\frac{12}{7}a^{12}-\frac{55}{7}a^{11}-\frac{96}{7}a^{10}+\frac{65}{7}a^{9}+\frac{178}{7}a^{8}+\frac{10}{7}a^{7}-\frac{228}{7}a^{6}-\frac{95}{7}a^{5}+\frac{152}{7}a^{4}+\frac{103}{7}a^{3}-\frac{73}{7}a^{2}-\frac{73}{7}a+\frac{38}{7}$, $\frac{19}{7}a^{13}+\frac{11}{7}a^{12}-\frac{51}{7}a^{11}-\frac{88}{7}a^{10}+\frac{66}{7}a^{9}+\frac{176}{7}a^{8}+\frac{1}{7}a^{7}-\frac{237}{7}a^{6}-\frac{90}{7}a^{5}+\frac{172}{7}a^{4}+\frac{109}{7}a^{3}-\frac{85}{7}a^{2}-\frac{78}{7}a+\frac{43}{7}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 14.647522546 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 14.647522546 \cdot 1}{2\cdot\sqrt{103886084489093}}\cr\approx \mathstrut & 0.17684577641 \end{aligned}\]
Galois group
$C_2^7.S_7$ (as 14T57):
A non-solvable group of order 645120 |
The 110 conjugacy class representatives for $C_2^7.S_7$ |
Character table for $C_2^7.S_7$ |
Intermediate fields
7.3.1400041.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 sibling: | data not computed |
Degree 28 siblings: | data not computed |
Degree 42 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | R | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(53\) | 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
53.10.0.1 | $x^{10} + x^{6} + x^{4} + 27 x^{3} + 15 x^{2} + 29 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(977\) | Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
\(1433\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |