Normalized defining polynomial
\( x^{14} - 14x^{12} + 168x^{10} - 1792x^{8} + 10976x^{6} - 12096x^{4} + 9856x^{2} - 5600 \)
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: | \(338637882455405716008075264\) \(\medspace = 2^{27}\cdot 3^{12}\cdot 7^{15}\) | 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: | \(78.52\) | 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))
| |
Galois root discriminant: | $2^{27/14}3^{6/7}7^{47/42}\approx 86.14376989441227$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{14}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{8}a^{9}$, $\frac{1}{8}a^{10}$, $\frac{1}{8}a^{11}$, $\frac{1}{157083551312}a^{12}-\frac{178197093}{39270887828}a^{10}-\frac{942432998}{9817721957}a^{8}-\frac{785391739}{9817721957}a^{6}+\frac{4074773933}{19635443914}a^{4}-\frac{942713421}{9817721957}a^{2}+\frac{2463001860}{9817721957}$, $\frac{1}{785417756560}a^{13}+\frac{2409881216}{49088609785}a^{11}-\frac{17357185941}{392708878280}a^{9}-\frac{12959288913}{196354439140}a^{7}+\frac{23710217847}{98177219570}a^{5}-\frac{942713421}{49088609785}a^{3}-\frac{17172442054}{49088609785}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{192327397}{39270887828}a^{12}+\frac{2512863669}{39270887828}a^{10}-\frac{15004184661}{19635443914}a^{8}+\frac{317027040011}{39270887828}a^{6}-\frac{455041470048}{9817721957}a^{4}+\frac{171606076480}{9817721957}a^{2}-\frac{357456736981}{9817721957}$, $\frac{361582385}{157083551312}a^{12}+\frac{2384222773}{78541775656}a^{10}-\frac{7119125699}{19635443914}a^{8}+\frac{150592048205}{39270887828}a^{6}-\frac{216860906016}{9817721957}a^{4}+\frac{88293572701}{9817721957}a^{2}-\frac{147931378323}{9817721957}$, $\frac{493612751}{78541775656}a^{12}+\frac{6494100669}{78541775656}a^{10}-\frac{38810815813}{39270887828}a^{8}+\frac{205237647517}{19635443914}a^{6}-\frac{590675935185}{9817721957}a^{4}+\frac{238807979321}{9817721957}a^{2}-\frac{398489941739}{9817721957}$, $\frac{36129}{33151180}a^{13}+\frac{257703}{16575590}a^{11}-\frac{1556473}{8287795}a^{9}+\frac{16695027}{8287795}a^{7}-\frac{104544426}{8287795}a^{5}+\frac{144258996}{8287795}a^{3}-\frac{190363656}{8287795}a+15$, $\frac{2377725111}{392708878280}a^{13}-\frac{129893339}{157083551312}a^{12}+\frac{34808519979}{392708878280}a^{11}+\frac{686049875}{39270887828}a^{10}-\frac{419858173693}{392708878280}a^{9}-\frac{2253592827}{9817721957}a^{8}+\frac{2249351255851}{196354439140}a^{7}+\frac{103861896767}{39270887828}a^{6}-\frac{3555420601422}{49088609785}a^{5}-\frac{407514991693}{19635443914}a^{4}+\frac{10228439476979}{98177219570}a^{3}+\frac{725589173918}{9817721957}a^{2}-\frac{2125337760747}{49088609785}a-\frac{483457380167}{9817721957}$, $\frac{88253914997}{785417756560}a^{13}+\frac{16386107207}{157083551312}a^{12}+\frac{579674767949}{392708878280}a^{11}-\frac{13456039692}{9817721957}a^{10}-\frac{1728153070867}{98177219570}a^{9}+\frac{641894552615}{39270887828}a^{8}+\frac{36552178036341}{196354439140}a^{7}-\frac{3394571596183}{19635443914}a^{6}-\frac{105303536636369}{98177219570}a^{5}+\frac{9780700464284}{9817721957}a^{4}+\frac{21272433112367}{49088609785}a^{3}-\frac{3954542595406}{9817721957}a^{2}-\frac{35913272702127}{49088609785}a+\frac{6656348244267}{9817721957}$, $\frac{38327159837}{392708878280}a^{13}+\frac{3471407567}{39270887828}a^{12}-\frac{504379047163}{392708878280}a^{11}-\frac{22862932839}{19635443914}a^{10}+\frac{6014847992691}{392708878280}a^{9}+\frac{136355620578}{9817721957}a^{8}-\frac{31811896729887}{196354439140}a^{7}-\frac{5769902288209}{39270887828}a^{6}+\frac{91796515196323}{98177219570}a^{5}+\frac{16669414410905}{19635443914}a^{4}-\frac{19340933215089}{49088609785}a^{3}-\frac{3647587722633}{9817721957}a^{2}+\frac{30546884074544}{49088609785}a+\frac{5828412211385}{9817721957}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 147637191.68695623 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 147637191.68695623 \cdot 1}{2\cdot\sqrt{338637882455405716008075264}}\cr\approx \mathstrut & 0.987272889594885 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times F_7$ (as 14T7):
A solvable group of order 84 |
The 14 conjugacy class representatives for $F_7 \times C_2$ |
Character table for $F_7 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{14}) \), 7.1.38423222208.4 |
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: | 14.0.48376840350772245144010752.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.27.121 | $x^{14} + 6 x^{12} + 4 x^{9} + 4 x^{7} + 4 x^{4} + 6$ | $14$ | $1$ | $27$ | $(C_7:C_3) \times C_2$ | $[3]_{7}^{3}$ |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.14.15.1 | $x^{14} + 14 x^{3} + 7 x^{2} + 21$ | $14$ | $1$ | $15$ | $F_7 \times C_2$ | $[7/6]_{6}^{2}$ |