Normalized defining polynomial
\( x^{14} - 522x^{7} - 324 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(48038057217218880000000\) \(\medspace = 2^{12}\cdot 3^{12}\cdot 5^{7}\cdot 7^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(41.70\) | 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: | $2^{6/7}3^{6/7}5^{1/2}7^{5/6}\approx 52.56784273085778$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{234}a^{7}+\frac{5}{13}$, $\frac{1}{234}a^{8}+\frac{5}{13}a$, $\frac{1}{234}a^{9}+\frac{5}{13}a^{2}$, $\frac{1}{234}a^{10}+\frac{5}{13}a^{3}$, $\frac{1}{702}a^{11}+\frac{5}{39}a^{4}$, $\frac{1}{4914}a^{12}-\frac{1}{1638}a^{11}-\frac{1}{1638}a^{10}+\frac{1}{1638}a^{9}+\frac{1}{819}a^{8}+\frac{1}{546}a^{7}+\frac{2}{21}a^{6}-\frac{8}{273}a^{5}-\frac{5}{91}a^{4}-\frac{44}{91}a^{3}+\frac{31}{91}a^{2}+\frac{36}{91}a-\frac{11}{91}$, $\frac{1}{4914}a^{13}+\frac{1}{2457}a^{11}-\frac{1}{819}a^{10}-\frac{1}{819}a^{9}+\frac{1}{819}a^{8}-\frac{1}{546}a^{7}-\frac{1}{13}a^{6}-\frac{1}{7}a^{5}-\frac{16}{273}a^{4}-\frac{10}{91}a^{3}+\frac{3}{91}a^{2}-\frac{29}{91}a+\frac{37}{91}$
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)
| |
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: | $\frac{1}{234}a^{7}-\frac{8}{13}$, $\frac{8}{2457}a^{13}-\frac{5}{4914}a^{12}-\frac{1}{546}a^{11}-\frac{1}{273}a^{10}+\frac{5}{1638}a^{9}+\frac{4}{819}a^{8}-\frac{466}{273}a^{6}+\frac{48}{91}a^{5}+\frac{89}{91}a^{4}+\frac{165}{91}a^{3}-\frac{170}{91}a^{2}-\frac{24}{13}a+\frac{11}{7}$, $\frac{1}{234}a^{11}+\frac{1}{117}a^{10}+\frac{1}{234}a^{9}-\frac{89}{39}a^{4}-\frac{55}{13}a^{3}-\frac{34}{13}a^{2}+1$, $\frac{1}{2457}a^{13}-\frac{1}{702}a^{12}+\frac{2}{2457}a^{11}-\frac{2}{819}a^{10}+\frac{1}{546}a^{9}+\frac{2}{819}a^{8}-\frac{3}{182}a^{7}-\frac{2}{13}a^{6}+\frac{160}{273}a^{5}-\frac{32}{273}a^{4}+\frac{71}{91}a^{3}-\frac{50}{91}a^{2}+\frac{33}{91}a-\frac{31}{91}$, $\frac{89}{2457}a^{13}+\frac{61}{1638}a^{12}+\frac{68}{2457}a^{11}+\frac{2}{117}a^{10}+\frac{1}{182}a^{9}-\frac{1}{126}a^{8}-\frac{3}{182}a^{7}-\frac{5168}{273}a^{6}-\frac{5312}{273}a^{5}-\frac{188}{13}a^{4}-\frac{809}{91}a^{3}-\frac{254}{91}a^{2}+\frac{30}{7}a+\frac{814}{91}$, $\frac{25}{819}a^{13}-\frac{29}{4914}a^{12}-\frac{20}{2457}a^{11}+\frac{22}{819}a^{10}-\frac{2}{117}a^{9}+\frac{2}{819}a^{8}+\frac{23}{1638}a^{7}-\frac{1453}{91}a^{6}+\frac{281}{91}a^{5}+\frac{1178}{273}a^{4}-\frac{184}{13}a^{3}+\frac{783}{91}a^{2}-\frac{32}{91}a-\frac{613}{91}$, $\frac{5}{819}a^{13}-\frac{22}{2457}a^{12}+\frac{11}{819}a^{11}-\frac{1}{819}a^{10}+\frac{11}{819}a^{9}-\frac{1}{234}a^{8}+\frac{37}{1638}a^{7}-\frac{288}{91}a^{6}+\frac{425}{91}a^{5}-\frac{1906}{273}a^{4}+\frac{68}{91}a^{3}-\frac{105}{13}a^{2}+\frac{17}{91}a-\frac{569}{91}$ (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: | \( 709188.6741093722 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 709188.6741093722 \cdot 1}{2\cdot\sqrt{48038057217218880000000}}\cr\approx \mathstrut & 0.398178933907917 \end{aligned}\] (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
\(\Q(\sqrt{5}) \), 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 | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.12.1 | $x^{14} + 7 x^{13} + 28 x^{12} + 77 x^{11} + 161 x^{10} + 266 x^{9} + 357 x^{8} + 397 x^{7} + 371 x^{6} + 224 x^{5} + 21 x^{4} + 7 x^{3} + 70 x^{2} + 35 x + 7$ | $7$ | $2$ | $12$ | $(C_7:C_3) \times C_2$ | $[\ ]_{7}^{6}$ |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |