Normalized defining polynomial
\( x^{14} - 7302x^{7} + 14823774 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-9877313453888738375765065728\) \(\medspace = -\,2^{20}\cdot 3^{12}\cdot 7^{10}\cdot 13^{7}\) | 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: | \(99.91\) | 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: | $2^{10/7}3^{6/7}7^{5/6}13^{1/2}\approx 125.95748174737544$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-13}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2373}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{7}a^{2}-\frac{2}{7}a+\frac{36}{113}$, $\frac{1}{2373}a^{8}-\frac{313}{791}a$, $\frac{1}{16611}a^{9}-\frac{1895}{5537}a^{2}$, $\frac{1}{116277}a^{10}+\frac{14716}{38759}a^{3}$, $\frac{1}{813939}a^{11}-\frac{24043}{271313}a^{4}$, $\frac{1}{5697573}a^{12}+\frac{518583}{1899191}a^{5}$, $\frac{1}{279181077}a^{13}-\frac{1}{13294337}a^{12}+\frac{2}{5697573}a^{11}+\frac{1}{813939}a^{10}-\frac{1}{38759}a^{9}+\frac{2}{16611}a^{8}-\frac{45062001}{93060359}a^{6}+\frac{6041015}{13294337}a^{5}-\frac{319399}{1899191}a^{4}+\frac{53475}{271313}a^{3}+\frac{5685}{38759}a^{2}-\frac{2208}{5537}a+\frac{2}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $6$ | 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)
| |
Fundamental units: | $\frac{95}{93060359}a^{13}-\frac{22}{39883011}a^{12}-\frac{20}{1899191}a^{11}+\frac{5}{813939}a^{10}-\frac{8}{116277}a^{9}+\frac{1}{5537}a^{8}-\frac{340743}{93060359}a^{6}-\frac{13680}{13294337}a^{5}+\frac{86015}{1899191}a^{4}-\frac{3938}{271313}a^{3}+\frac{15160}{38759}a^{2}-\frac{3312}{5537}a-\frac{11}{7}$, $\frac{190}{279181077}a^{13}+\frac{11}{39883011}a^{12}-\frac{4}{1899191}a^{11}+\frac{5}{271313}a^{10}-\frac{1}{38759}a^{9}+\frac{2}{16611}a^{8}-\frac{227162}{93060359}a^{6}+\frac{6840}{13294337}a^{5}+\frac{17203}{1899191}a^{4}-\frac{11814}{271313}a^{3}+\frac{5685}{38759}a^{2}-\frac{2208}{5537}a-\frac{5}{7}$, $\frac{95}{39883011}a^{13}+\frac{4}{271313}a^{11}+\frac{1}{2373}a^{8}-\frac{113581}{13294337}a^{6}-\frac{17203}{271313}a^{4}-\frac{1104}{791}a+1$, $\frac{106964375965}{279181077}a^{13}+\frac{49515989260}{39883011}a^{12}+\frac{22872651814}{5697573}a^{11}+\frac{10542443981}{813939}a^{10}+\frac{4848467410}{116277}a^{9}+\frac{2224775243}{16611}a^{8}+\frac{339500303}{791}a^{7}-\frac{132658898326740}{93060359}a^{6}-\frac{62347703647202}{13294337}a^{5}-\frac{29237006478964}{1899191}a^{4}-\frac{13680168385263}{271313}a^{3}-\frac{6387183335492}{38759}a^{2}-\frac{2975745746215}{5537}a-\frac{1383425605235}{791}$, $\frac{21143574811}{279181077}a^{13}+\frac{9393730613}{39883011}a^{12}+\frac{557104153}{5697573}a^{11}-\frac{1723876733}{813939}a^{10}-\frac{1060448185}{116277}a^{9}-\frac{205949453}{16611}a^{8}+\frac{38867642}{791}a^{7}-\frac{21802314042064}{93060359}a^{6}-\frac{13601498435656}{13294337}a^{5}-\frac{2709570499950}{1899191}a^{4}+\frac{1458704401172}{271313}a^{3}+\frac{1380569316142}{38759}a^{2}+\frac{437714431316}{5537}a-\frac{59320399225}{791}$, $\frac{57246116498}{1899191}a^{13}-\frac{369884629221}{1899191}a^{12}-\frac{312958706708}{271313}a^{11}-\frac{112013498630}{38759}a^{10}-\frac{1161061708}{5537}a^{9}+\frac{23447930416}{791}a^{8}+\frac{14625052212}{113}a^{7}+\frac{41673139072736}{1899191}a^{6}+\frac{20\!\cdots\!26}{1899191}a^{5}+\frac{12\!\cdots\!18}{271313}a^{4}+\frac{291378605805306}{38759}a^{3}-\frac{81436165025122}{5537}a^{2}-\frac{112604933997144}{791}a-\frac{51397870705555}{113}$ (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: | \( 151362953.43555072 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 151362953.43555072 \cdot 4}{2\cdot\sqrt{9877313453888738375765065728}}\cr\approx \mathstrut & 1.17757682290025 \end{aligned}\] (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
\(\Q(\sqrt{-13}) \), 7.1.784147392.1 |
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: | 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 | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.20.10 | $x^{14} + 2 x^{13} + 2 x^{10} + 2 x^{7} + 6$ | $14$ | $1$ | $20$ | $(C_7:C_3) \times C_2$ | $[2]_{7}^{3}$ |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |