Normalized defining polynomial
\( x^{14} + 1406250 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-1574111058893828259840000000\)
\(\medspace = -\,2^{27}\cdot 3^{12}\cdot 5^{7}\cdot 7^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(87.63\) | 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: | $2^{27/14}3^{6/7}5^{1/2}7^{5/6}\approx 110.4720194582544$ | ||
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: | \(\Q(\sqrt{-10}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{25}a^{4}$, $\frac{1}{25}a^{5}$, $\frac{1}{125}a^{6}$, $\frac{1}{375}a^{7}$, $\frac{1}{1875}a^{8}$, $\frac{1}{1875}a^{9}$, $\frac{1}{9375}a^{10}$, $\frac{1}{9375}a^{11}$, $\frac{1}{328125}a^{12}+\frac{1}{21875}a^{10}+\frac{2}{13125}a^{8}+\frac{2}{875}a^{6}-\frac{1}{175}a^{4}-\frac{3}{35}a^{2}-\frac{2}{7}$, $\frac{1}{328125}a^{13}+\frac{1}{21875}a^{11}+\frac{2}{13125}a^{9}-\frac{1}{2625}a^{7}-\frac{1}{175}a^{5}-\frac{3}{35}a^{3}-\frac{2}{7}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $6$ | 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: |
$\frac{2}{328125}a^{12}-\frac{1}{65625}a^{10}-\frac{1}{4375}a^{8}-\frac{3}{875}a^{6}+\frac{1}{35}a^{4}+\frac{8}{35}a^{2}-\frac{11}{7}$, $\frac{1}{328125}a^{12}+\frac{1}{21875}a^{10}+\frac{2}{13125}a^{8}+\frac{2}{875}a^{6}-\frac{1}{175}a^{4}-\frac{3}{35}a^{2}+\frac{5}{7}$, $\frac{1}{1875}a^{8}+\frac{1}{125}a^{6}+\frac{1}{25}a^{4}-1$, $\frac{144}{109375}a^{13}+\frac{1252}{328125}a^{12}+\frac{229}{21875}a^{11}+\frac{1817}{65625}a^{10}+\frac{899}{13125}a^{9}+\frac{2056}{13125}a^{8}+\frac{58}{175}a^{7}+\frac{509}{875}a^{6}+\frac{4}{7}a^{5}-\frac{202}{175}a^{4}-\frac{372}{35}a^{3}-\frac{1726}{35}a^{2}-\frac{1263}{7}a-\frac{4163}{7}$, $\frac{18898703}{328125}a^{13}-\frac{71475409}{65625}a^{12}+\frac{141860384}{65625}a^{11}+\frac{201337411}{65625}a^{10}-\frac{310022458}{13125}a^{9}+\frac{434011108}{13125}a^{8}+\frac{17459892}{175}a^{7}-\frac{426877949}{875}a^{6}+\frac{71509321}{175}a^{5}+\frac{475460507}{175}a^{4}-\frac{335372836}{35}a^{3}+\frac{82112504}{35}a^{2}+\frac{467808660}{7}a-\frac{1241663303}{7}$, $\frac{34\!\cdots\!73}{9375}a^{13}-\frac{34\!\cdots\!13}{109375}a^{12}+\frac{10\!\cdots\!07}{9375}a^{11}-\frac{14\!\cdots\!86}{65625}a^{10}+\frac{28\!\cdots\!61}{1875}a^{9}+\frac{13\!\cdots\!97}{13125}a^{8}-\frac{69\!\cdots\!26}{125}a^{7}+\frac{14\!\cdots\!48}{875}a^{6}-\frac{68\!\cdots\!19}{25}a^{5}-\frac{80\!\cdots\!62}{175}a^{4}+22\!\cdots\!40a^{3}-\frac{32\!\cdots\!16}{35}a^{2}+23\!\cdots\!27a-\frac{20\!\cdots\!05}{7}$
| 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: | \( 132394727.38181365 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 132394727.38181365 \cdot 2}{2\cdot\sqrt{1574111058893828259840000000}}\cr\approx \mathstrut & 1.29006645658035 \end{aligned}\] (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
\(\Q(\sqrt{-10}) \), 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 | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.14.27.21 | $x^{14} + 4 x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 14$ | $14$ | $1$ | $27$ | $(C_7:C_3) \times C_2$ | $[3]_{7}^{3}$ |
\(3\)
| 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | 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 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |