Normalized defining polynomial
\( x^{14} + x^{12} - 3x^{10} - 2x^{8} + 5x^{6} + 2x^{4} - 2x^{2} - 1 \)
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: | \(511335829405696\) \(\medspace = 2^{16}\cdot 19^{2}\cdot 4649^{2}\) | 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: | \(11.24\) | 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: | $2^{5/2}19^{1/2}4649^{1/2}\approx 1681.2471561314237$ | ||
Ramified primes: | \(2\), \(19\), \(4649\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$
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: | $a$, $a^{13}-4a^{9}+a^{7}+6a^{5}-3a^{3}-2a$, $a^{13}+a^{12}+\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-3a^{9}-\frac{7}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+4a^{5}+\frac{11}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}$, $\frac{1}{2}a^{13}-a^{12}+\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-a^{9}+3a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+a^{5}-5a^{4}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}a+2$, $a^{10}+a^{8}-2a^{6}-a^{4}+3a^{2}+1$, $\frac{3}{2}a^{12}-\frac{1}{2}a^{11}+a^{10}-\frac{1}{2}a^{9}-\frac{9}{2}a^{8}+\frac{3}{2}a^{7}-a^{6}+\frac{1}{2}a^{5}+\frac{15}{2}a^{4}-\frac{5}{2}a^{3}+\frac{1}{2}a-2$, $\frac{1}{2}a^{13}-2a^{9}-\frac{1}{2}a^{8}+3a^{5}+\frac{3}{2}a^{4}-a^{3}-\frac{3}{2}a-\frac{3}{2}$ | 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: | \( 44.0785273821 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 44.0785273821 \cdot 1}{2\cdot\sqrt{511335829405696}}\cr\approx \mathstrut & 0.239873996071 \end{aligned}\]
Galois group
$C_2^6.S_7$ (as 14T55):
A non-solvable group of order 322560 |
The 55 conjugacy class representatives for $C_2^6.S_7$ are not computed |
Character table for $C_2^6.S_7$ is not computed |
Intermediate fields
7.1.1413296.1 |
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 sibling: | data not computed |
Degree 42 siblings: | data not computed |
Minimal sibling: | 14.0.2749615457296632415053253373527995908096.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
2.8.16.37 | $x^{8} + 8 x^{6} - 4 x^{5} + 28 x^{4} + 8 x^{3} + 88 x^{2} + 72 x + 108$ | $4$ | $2$ | $16$ | $C_8:C_2$ | $[2, 3, 3]^{2}$ | |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(4649\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |