Normalized defining polynomial
\( x^{14} - 3 x^{12} - 2 x^{11} + 5 x^{10} + 4 x^{9} - 6 x^{8} - 7 x^{7} + 7 x^{6} + 9 x^{5} - 4 x^{4} + \cdots - 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-411650990546875\) \(\medspace = -\,5^{7}\cdot 24851\cdot 212029\) | 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: | \(11.06\) | 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: | $5^{1/2}24851^{1/2}212029^{1/2}\approx 162313.47262319294$ | ||
Ramified primes: | \(5\), \(24851\), \(212029\) | 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: | $\Q(\sqrt{-26345663395}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}-\frac{3}{7}a^{12}-\frac{1}{7}a^{11}+\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{2}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{2}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $a$, $\frac{1}{7}a^{13}+\frac{4}{7}a^{12}-\frac{8}{7}a^{11}-\frac{13}{7}a^{10}+\frac{9}{7}a^{9}+\frac{26}{7}a^{8}-a^{7}-5a^{6}+a^{5}+\frac{44}{7}a^{4}+\frac{4}{7}a^{3}-\frac{32}{7}a^{2}+\frac{2}{7}$, $\frac{6}{7}a^{13}+\frac{3}{7}a^{12}-\frac{13}{7}a^{11}-\frac{15}{7}a^{10}+\frac{12}{7}a^{9}+\frac{16}{7}a^{8}-2a^{7}-4a^{6}+2a^{5}+\frac{26}{7}a^{4}+\frac{3}{7}a^{3}+\frac{4}{7}a^{2}+a-\frac{2}{7}$, $\frac{6}{7}a^{13}+\frac{3}{7}a^{12}-\frac{13}{7}a^{11}-\frac{15}{7}a^{10}+\frac{12}{7}a^{9}+\frac{16}{7}a^{8}-2a^{7}-4a^{6}+2a^{5}+\frac{26}{7}a^{4}+\frac{3}{7}a^{3}+\frac{4}{7}a^{2}+a-\frac{9}{7}$, $\frac{6}{7}a^{13}+\frac{10}{7}a^{12}-\frac{20}{7}a^{11}-\frac{36}{7}a^{10}+\frac{12}{7}a^{9}+\frac{58}{7}a^{8}-a^{7}-11a^{6}-a^{5}+\frac{96}{7}a^{4}+\frac{38}{7}a^{3}-\frac{38}{7}a^{2}-2a+\frac{5}{7}$, $\frac{5}{7}a^{13}-\frac{1}{7}a^{12}-\frac{12}{7}a^{11}-\frac{9}{7}a^{10}+\frac{17}{7}a^{9}+\frac{18}{7}a^{8}-3a^{7}-4a^{6}+3a^{5}+\frac{38}{7}a^{4}-\frac{1}{7}a^{3}-\frac{20}{7}a^{2}-a+\frac{10}{7}$, $\frac{1}{7}a^{13}-\frac{3}{7}a^{12}-\frac{1}{7}a^{11}+\frac{8}{7}a^{10}+\frac{2}{7}a^{9}-\frac{16}{7}a^{8}-a^{7}+3a^{6}+2a^{5}-\frac{26}{7}a^{4}-\frac{10}{7}a^{3}+\frac{24}{7}a^{2}+2a-\frac{12}{7}$, $\frac{3}{7}a^{13}+\frac{5}{7}a^{12}-\frac{3}{7}a^{11}-\frac{18}{7}a^{10}-\frac{8}{7}a^{9}+\frac{22}{7}a^{8}+2a^{7}-4a^{6}-4a^{5}+\frac{27}{7}a^{4}+\frac{40}{7}a^{3}+\frac{2}{7}a^{2}-2a+\frac{6}{7}$ | 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: | \( 42.0905477867 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 42.0905477867 \cdot 1}{2\cdot\sqrt{411650990546875}}\cr\approx \mathstrut & 0.162520939525 \end{aligned}\]
Galois group
$S_7\wr C_2$ (as 14T61):
A non-solvable group of order 50803200 |
The 135 conjugacy class representatives for $S_7\wr C_2$ |
Character table for $S_7\wr C_2$ |
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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.14.0.1}{14} }$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.14.7.1 | $x^{14} + 140 x^{13} + 8435 x^{12} + 284200 x^{11} + 5810525 x^{10} + 72852500 x^{9} + 534104381 x^{8} + 1994350486 x^{7} + 2670547075 x^{6} + 1822151870 x^{5} + 743294125 x^{4} + 386790250 x^{3} + 1508497384 x^{2} + 5074882448 x + 4401772109$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
\(24851\) | $\Q_{24851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{24851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
\(212029\) | $\Q_{212029}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |