Normalized defining polynomial
\( x^{14} - 14x^{12} + 55x^{10} - 48x^{8} - 107x^{6} + 198x^{4} - 103x^{2} + 17 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-98547323936652881051648\) \(\medspace = -\,2^{14}\cdot 17\cdot 29^{12}\) | 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: | \(43.89\) | 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: | $2^{127/64}17^{1/2}29^{6/7}\approx 292.4589096608059$ | ||
Ramified primes: | \(2\), \(17\), \(29\) | 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: | \(\Q(\sqrt{-17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2419}a^{12}+\frac{1026}{2419}a^{10}+\frac{316}{2419}a^{8}-\frac{392}{2419}a^{6}+\frac{1024}{2419}a^{4}+\frac{798}{2419}a^{2}+\frac{100}{2419}$, $\frac{1}{2419}a^{13}+\frac{1026}{2419}a^{11}+\frac{316}{2419}a^{9}-\frac{392}{2419}a^{7}+\frac{1024}{2419}a^{5}+\frac{798}{2419}a^{3}+\frac{100}{2419}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $12$ | 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: | $\frac{535}{2419}a^{12}-\frac{7460}{2419}a^{10}+\frac{28758}{2419}a^{8}-\frac{21038}{2419}a^{6}-\frac{66586}{2419}a^{4}+\frac{97946}{2419}a^{2}-\frac{26327}{2419}$, $\frac{980}{2419}a^{12}-\frac{12919}{2419}a^{10}+\frac{43590}{2419}a^{8}-\frac{14053}{2419}a^{6}-\frac{111639}{2419}a^{4}+\frac{107139}{2419}a^{2}-\frac{25369}{2419}$, $\frac{56}{2419}a^{12}-\frac{600}{2419}a^{10}+\frac{763}{2419}a^{8}+\frac{4657}{2419}a^{6}-\frac{10388}{2419}a^{4}+\frac{3565}{2419}a^{2}+\frac{762}{2419}$, $\frac{1964}{2419}a^{12}-\frac{26572}{2419}a^{10}+\frac{95701}{2419}a^{8}-\frac{51445}{2419}a^{6}-\frac{228858}{2419}a^{4}+\frac{285202}{2419}a^{2}-\frac{81785}{2419}$, $\frac{370}{2419}a^{12}-\frac{5001}{2419}a^{10}+\frac{17741}{2419}a^{8}-\frac{7157}{2419}a^{6}-\frac{46864}{2419}a^{4}+\frac{48522}{2419}a^{2}-\frac{8961}{2419}$, $\frac{535}{2419}a^{12}-\frac{7460}{2419}a^{10}+\frac{28758}{2419}a^{8}-\frac{21038}{2419}a^{6}-\frac{66586}{2419}a^{4}+\frac{100365}{2419}a^{2}-\frac{31165}{2419}$, $\frac{1070}{2419}a^{13}-\frac{1070}{2419}a^{12}-\frac{14920}{2419}a^{11}+\frac{14920}{2419}a^{10}+\frac{57516}{2419}a^{9}-\frac{57516}{2419}a^{8}-\frac{42076}{2419}a^{7}+\frac{42076}{2419}a^{6}-\frac{133172}{2419}a^{5}+\frac{133172}{2419}a^{4}+\frac{198311}{2419}a^{3}-\frac{198311}{2419}a^{2}-\frac{55073}{2419}a+\frac{55073}{2419}$, $\frac{370}{2419}a^{12}-\frac{5001}{2419}a^{10}+\frac{17741}{2419}a^{8}-\frac{7157}{2419}a^{6}-\frac{46864}{2419}a^{4}+\frac{48522}{2419}a^{2}-a-\frac{11380}{2419}$, $\frac{3374}{2419}a^{13}-\frac{1964}{2419}a^{12}-\frac{45826}{2419}a^{11}+\frac{26572}{2419}a^{10}+\frac{166316}{2419}a^{9}-\frac{95701}{2419}a^{8}-\frac{91337}{2419}a^{7}+\frac{51445}{2419}a^{6}-\frac{400910}{2419}a^{5}+\frac{228858}{2419}a^{4}+\frac{496000}{2419}a^{3}-\frac{285202}{2419}a^{2}-\frac{136724}{2419}a+\frac{79366}{2419}$, $\frac{3744}{2419}a^{13}+\frac{2155}{2419}a^{12}-\frac{50827}{2419}a^{11}-\frac{28964}{2419}a^{10}+\frac{184057}{2419}a^{9}+\frac{102839}{2419}a^{8}-\frac{98494}{2419}a^{7}-\frac{51328}{2419}a^{6}-\frac{447774}{2419}a^{5}-\frac{250984}{2419}a^{4}+\frac{544522}{2419}a^{3}+\frac{299737}{2419}a^{2}-\frac{140847}{2419}a-\frac{74780}{2419}$, $\frac{5503}{2419}a^{13}-\frac{24}{59}a^{12}-\frac{74857}{2419}a^{11}+\frac{333}{59}a^{10}+\frac{273034}{2419}a^{9}-\frac{1271}{59}a^{8}-\frac{156663}{2419}a^{7}+\frac{912}{59}a^{6}-\frac{649490}{2419}a^{5}+\frac{2859}{59}a^{4}+\frac{833045}{2419}a^{3}-\frac{4284}{59}a^{2}-\frac{238294}{2419}a+\frac{1435}{59}$, $\frac{1429}{2419}a^{13}-\frac{605}{2419}a^{12}-\frac{19112}{2419}a^{11}+\frac{8210}{2419}a^{10}+\frac{66943}{2419}a^{9}-\frac{29107}{2419}a^{8}-\frac{30407}{2419}a^{7}+\frac{9774}{2419}a^{6}-\frac{162272}{2419}a^{5}+\frac{79571}{2419}a^{4}+\frac{187256}{2419}a^{3}-\frac{66722}{2419}a^{2}-\frac{55458}{2419}a+\frac{9651}{2419}$ | 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: | \( 4761814.27223 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{1}\cdot 4761814.27223 \cdot 1}{2\cdot\sqrt{98547323936652881051648}}\cr\approx \mathstrut & 0.195191029775 \end{aligned}\]
Galois group
$C_2\wr C_7$ (as 14T29):
A solvable group of order 896 |
The 32 conjugacy class representatives for $C_2 \wr C_7$ |
Character table for $C_2 \wr C_7$ |
Intermediate fields
7.7.594823321.1 |
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 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 | R | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.14.4 | $x^{14} + 2 x^{13} + 38 x^{12} + 240 x^{11} + 884 x^{10} + 4840 x^{9} + 13432 x^{8} + 46528 x^{7} + 102384 x^{6} + 227552 x^{5} + 361632 x^{4} + 465408 x^{3} + 432832 x^{2} + 199040 x + 22656$ | $2$ | $7$ | $14$ | $C_2 \wr C_7$ | $[2, 2, 2, 2, 2, 2, 2]^{7}$ |
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(29\) | 29.14.12.1 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |