Normalized defining polynomial
\( x^{14} - 10338x^{7} - 1406250 \)
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: | \(3067497137765510214413451264\) \(\medspace = 2^{20}\cdot 3^{12}\cdot 7^{10}\cdot 11^{7}\) | 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: | \(91.91\) | 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))
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Galois root discriminant: | $2^{10/7}3^{6/7}7^{5/6}11^{1/2}\approx 115.86403148304235$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(11\) | 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: | \(\Q(\sqrt{11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1599}a^{7}-\frac{124}{533}$, $\frac{1}{7995}a^{8}+\frac{409}{2665}a$, $\frac{1}{39975}a^{9}-\frac{4921}{13325}a^{2}$, $\frac{1}{199875}a^{10}-\frac{31571}{66625}a^{3}$, $\frac{1}{999375}a^{11}-\frac{31571}{333125}a^{4}$, $\frac{1}{34978125}a^{12}-\frac{1}{2331875}a^{11}-\frac{1}{279825}a^{9}+\frac{1}{18655}a^{8}-\frac{3}{7}a^{6}-\frac{480403}{1665625}a^{5}-\frac{571537}{2331875}a^{4}-\frac{3}{7}a^{3}+\frac{703}{13325}a^{2}-\frac{6768}{18655}a-\frac{3}{7}$, $\frac{1}{174890625}a^{13}-\frac{2}{6995625}a^{11}-\frac{1}{1399125}a^{10}+\frac{2}{55965}a^{8}+\frac{1}{11193}a^{7}-\frac{20019071}{58296875}a^{6}+\frac{2}{7}a^{5}-\frac{38569}{333125}a^{4}-\frac{35054}{466375}a^{3}+\frac{2}{7}a^{2}+\frac{1259}{2665}a+\frac{1475}{3731}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{14}$, which has order $14$ (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: | $\frac{17}{8328125}a^{13}-\frac{118}{34978125}a^{12}+\frac{4}{466375}a^{11}-\frac{1}{199875}a^{10}-\frac{8}{279825}a^{9}+\frac{1}{18655}a^{8}-\frac{1277097}{58296875}a^{6}+\frac{56304}{1665625}a^{5}-\frac{45727}{466375}a^{4}+\frac{21122}{466375}a^{3}+\frac{5624}{13325}a^{2}-\frac{6768}{18655}a+\frac{11}{7}$, $\frac{119}{24984375}a^{13}-\frac{4}{333125}a^{11}+\frac{1}{7995}a^{8}-\frac{425699}{8328125}a^{6}+\frac{45727}{333125}a^{4}-\frac{2256}{2665}a-1$, $\frac{34}{24984375}a^{13}+\frac{59}{34978125}a^{12}+\frac{4}{2331875}a^{11}-\frac{1}{66625}a^{10}-\frac{1}{93275}a^{9}+\frac{2}{55965}a^{8}-\frac{851398}{58296875}a^{6}-\frac{28152}{1665625}a^{5}-\frac{45727}{2331875}a^{4}+\frac{63366}{466375}a^{3}+\frac{2109}{13325}a^{2}-\frac{4512}{18655}a+\frac{5}{7}$, $\frac{1}{533}a^{7}-\frac{10499}{533}$, $\frac{61}{4996875}a^{13}+\frac{46}{999375}a^{12}+\frac{19}{333125}a^{11}-\frac{7}{199875}a^{10}+\frac{1}{13325}a^{9}+\frac{7}{2665}a^{8}-\frac{3}{533}a^{7}-\frac{260206}{1665625}a^{6}-\frac{119766}{333125}a^{5}-\frac{133922}{333125}a^{4}-\frac{112128}{66625}a^{3}-\frac{1438}{13325}a^{2}+\frac{594}{2665}a+\frac{1649}{533}$, $\frac{11821}{58296875}a^{13}+\frac{4512}{11659375}a^{12}+\frac{311}{538125}a^{11}+\frac{496}{1399125}a^{10}-\frac{33}{18655}a^{9}-\frac{38}{4305}a^{8}-\frac{72}{3731}a^{7}-\frac{121892998}{58296875}a^{6}-\frac{44251306}{11659375}a^{5}-\frac{952331}{179375}a^{4}-\frac{1801216}{466375}a^{3}+\frac{138064}{18655}a^{2}+\frac{60923}{1435}a+\frac{464377}{3731}$, $\frac{89522}{174890625}a^{13}+\frac{656}{853125}a^{12}+\frac{414}{2331875}a^{11}-\frac{2123}{466375}a^{10}-\frac{2271}{93275}a^{9}-\frac{1828}{18655}a^{8}-\frac{1349}{3731}a^{7}-\frac{384492812}{58296875}a^{6}-\frac{3607451}{284375}a^{5}-\frac{44874307}{2331875}a^{4}-\frac{8193426}{466375}a^{3}+\frac{947148}{93275}a^{2}+\frac{2005054}{18655}a+\frac{1265617}{3731}$ (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: | \( 40718019.19045277 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 40718019.19045277 \cdot 14}{2\cdot\sqrt{3067497137765510214413451264}}\cr\approx \mathstrut & 1.26657759434843 \end{aligned}\] (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
\(\Q(\sqrt{11}) \), 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 | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.20.10 | $x^{14} + 2 x^{13} + 2 x^{10} + 2 x^{7} + 6$ | $14$ | $1$ | $20$ | $(C_7:C_3) \times C_2$ | $[2]_{7}^{3}$ |
\(3\) | 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 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |