Normalized defining polynomial
\( x^{14} - x^{13} - 4 x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{9} + 2 x^{8} - 12 x^{7} + x^{5} + 2 x^{4} + 6 x^{3} - 3 x^{2} + x - 1 \)
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: | \(-334624313614643\) \(\medspace = -\,13^{4}\cdot 83\cdot 109^{4}\) | 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: | \(10.90\) | 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: | $13^{1/2}83^{1/2}109^{1/2}\approx 342.94460194031336$ | ||
Ramified primes: | \(13\), \(83\), \(109\) | 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: | \(\Q(\sqrt{-83}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{83}a^{13}+\frac{41}{83}a^{12}-\frac{25}{83}a^{11}+\frac{32}{83}a^{10}+\frac{19}{83}a^{9}-\frac{29}{83}a^{8}+\frac{29}{83}a^{7}-\frac{39}{83}a^{6}+\frac{22}{83}a^{5}+\frac{12}{83}a^{4}+\frac{8}{83}a^{3}+\frac{10}{83}a^{2}+\frac{2}{83}a+\frac{2}{83}$
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: | $\frac{8}{83}a^{13}-\frac{4}{83}a^{12}-\frac{34}{83}a^{11}+\frac{7}{83}a^{10}+\frac{69}{83}a^{9}+\frac{17}{83}a^{8}-\frac{100}{83}a^{7}-\frac{63}{83}a^{6}+\frac{10}{83}a^{5}+\frac{179}{83}a^{4}+\frac{147}{83}a^{3}-\frac{169}{83}a^{2}-\frac{67}{83}a-\frac{67}{83}$, $a$, $\frac{36}{83}a^{13}-\frac{18}{83}a^{12}-\frac{153}{83}a^{11}+\frac{73}{83}a^{10}+\frac{103}{83}a^{9}+\frac{35}{83}a^{8}+\frac{214}{83}a^{7}-\frac{325}{83}a^{6}-\frac{121}{83}a^{5}+\frac{100}{83}a^{4}-\frac{127}{83}a^{3}+\frac{277}{83}a^{2}-\frac{11}{83}a-\frac{11}{83}$, $\frac{12}{83}a^{13}-\frac{6}{83}a^{12}-\frac{51}{83}a^{11}+\frac{52}{83}a^{10}+\frac{62}{83}a^{9}-\frac{99}{83}a^{8}-\frac{67}{83}a^{7}-\frac{53}{83}a^{6}+\frac{181}{83}a^{5}+\frac{227}{83}a^{4}-\frac{70}{83}a^{3}-\frac{46}{83}a^{2}-\frac{142}{83}a+\frac{24}{83}$, $\frac{5}{83}a^{13}+\frac{39}{83}a^{12}-\frac{42}{83}a^{11}-\frac{172}{83}a^{10}+\frac{95}{83}a^{9}+\frac{187}{83}a^{8}+\frac{62}{83}a^{7}+\frac{54}{83}a^{6}-\frac{388}{83}a^{5}-\frac{106}{83}a^{4}+\frac{206}{83}a^{3}+\frac{50}{83}a^{2}+\frac{176}{83}a+\frac{10}{83}$, $\frac{39}{83}a^{13}-\frac{61}{83}a^{12}-\frac{145}{83}a^{11}+\frac{252}{83}a^{10}+\frac{77}{83}a^{9}-\frac{135}{83}a^{8}+\frac{52}{83}a^{7}-\frac{442}{83}a^{6}+\frac{277}{83}a^{5}+\frac{302}{83}a^{4}-\frac{186}{83}a^{3}+\frac{224}{83}a^{2}-\frac{254}{83}a-\frac{5}{83}$, $\frac{3}{83}a^{13}-\frac{43}{83}a^{12}+\frac{8}{83}a^{11}+\frac{179}{83}a^{10}-\frac{26}{83}a^{9}-\frac{170}{83}a^{8}-\frac{162}{83}a^{7}-\frac{117}{83}a^{6}+\frac{398}{83}a^{5}+\frac{202}{83}a^{4}-\frac{59}{83}a^{3}-\frac{53}{83}a^{2}-\frac{243}{83}a+\frac{6}{83}$, $\frac{8}{83}a^{13}-\frac{4}{83}a^{12}-\frac{34}{83}a^{11}+\frac{7}{83}a^{10}+\frac{69}{83}a^{9}+\frac{17}{83}a^{8}-\frac{100}{83}a^{7}-\frac{63}{83}a^{6}+\frac{10}{83}a^{5}+\frac{96}{83}a^{4}+\frac{147}{83}a^{3}-\frac{3}{83}a^{2}-\frac{67}{83}a-\frac{67}{83}$ | 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: | \( 37.8345982696 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 37.8345982696 \cdot 1}{2\cdot\sqrt{334624313614643}}\cr\approx \mathstrut & 0.162031595726 \end{aligned}\]
Galois group
$C_2^7.\GL(3,2)$ (as 14T51):
A non-solvable group of order 21504 |
The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$ |
Character table for $C_2^7.\GL(3,2)$ is not computed |
Intermediate fields
7.3.2007889.2 |
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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(83\) | 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.8.4.1 | $x^{8} + 28776 x^{7} + 310522274 x^{6} + 1489272514288 x^{5} + 2678510521605233 x^{4} + 178743008720712 x^{3} + 29612720181709536 x^{2} + 263388846138180416 x + 20054316486246464$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |