Normalized defining polynomial
\( x^{14} - 6 x^{13} + 16 x^{12} - 27 x^{11} + 36 x^{10} - 40 x^{9} + 37 x^{8} - 32 x^{7} + 20 x^{6} + \cdots + 1 \)
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: | \(-1092568542042991\) \(\medspace = -\,13^{4}\cdot 109^{4}\cdot 271\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.86\) | 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: | $13^{1/2}109^{1/2}271^{1/2}\approx 619.6829834681602$ | ||
Ramified primes: | \(13\), \(109\), \(271\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-271}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3929}a^{13}-\frac{1526}{3929}a^{12}+\frac{1426}{3929}a^{11}+\frac{1261}{3929}a^{10}+\frac{668}{3929}a^{9}-\frac{1718}{3929}a^{8}-\frac{1388}{3929}a^{7}-\frac{145}{3929}a^{6}+\frac{396}{3929}a^{5}-\frac{786}{3929}a^{4}+\frac{305}{3929}a^{3}+\frac{18}{3929}a^{2}+\frac{143}{3929}a-\frac{1264}{3929}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{272}{3929}a^{13}-\frac{2527}{3929}a^{12}+\frac{10688}{3929}a^{11}-\frac{26334}{3929}a^{10}+\frac{44181}{3929}a^{9}-\frac{58680}{3929}a^{8}+\frac{66441}{3929}a^{7}-\frac{63014}{3929}a^{6}+\frac{52706}{3929}a^{5}-\frac{33058}{3929}a^{4}+\frac{4380}{3929}a^{3}+\frac{967}{3929}a^{2}+\frac{3535}{3929}a+\frac{1944}{3929}$, $a$, $\frac{41}{3929}a^{13}+\frac{298}{3929}a^{12}-\frac{469}{3929}a^{11}-\frac{3305}{3929}a^{10}+\frac{11672}{3929}a^{9}-\frac{19361}{3929}a^{8}+\frac{25601}{3929}a^{7}-\frac{25590}{3929}a^{6}+\frac{20165}{3929}a^{5}-\frac{20439}{3929}a^{4}+\frac{8576}{3929}a^{3}+\frac{4667}{3929}a^{2}+\frac{1934}{3929}a-\frac{4676}{3929}$, $\frac{2115}{3929}a^{13}-\frac{9639}{3929}a^{12}+\frac{18163}{3929}a^{11}-\frac{20421}{3929}a^{10}+\frac{18025}{3929}a^{9}-\frac{7103}{3929}a^{8}-\frac{4586}{3929}a^{7}+\frac{7645}{3929}a^{6}-\frac{22911}{3929}a^{5}+\frac{23151}{3929}a^{4}+\frac{719}{3929}a^{3}-\frac{1220}{3929}a^{2}-\frac{88}{3929}a-\frac{1640}{3929}$, $\frac{718}{3929}a^{13}-\frac{3406}{3929}a^{12}+\frac{6257}{3929}a^{11}-\frac{6130}{3929}a^{10}+\frac{4215}{3929}a^{9}+\frac{182}{3929}a^{8}-\frac{6476}{3929}a^{7}+\frac{9831}{3929}a^{6}-\frac{18205}{3929}a^{5}+\frac{21073}{3929}a^{4}-\frac{8892}{3929}a^{3}+\frac{5066}{3929}a^{2}-\frac{7338}{3929}a+\frac{47}{3929}$, $\frac{925}{3929}a^{13}-\frac{4968}{3929}a^{12}+\frac{10693}{3929}a^{11}-\frac{12275}{3929}a^{10}+\frac{8905}{3929}a^{9}-\frac{1834}{3929}a^{8}-\frac{6975}{3929}a^{7}+\frac{11248}{3929}a^{6}-\frac{18742}{3929}a^{5}+\frac{23389}{3929}a^{4}-\frac{8621}{3929}a^{3}+\frac{934}{3929}a^{2}-\frac{1311}{3929}a+\frac{1642}{3929}$ | 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: | \( 35.5450250635 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 35.5450250635 \cdot 1}{2\cdot\sqrt{1092568542042991}}\cr\approx \mathstrut & 0.207866049190 \end{aligned}\]
Galois group
$C_2^7.\GL(3,2)$ (as 14T51):
A non-solvable group of order 21504 |
The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$ |
Character table for $C_2^7.\GL(3,2)$ is not computed |
Intermediate fields
7.3.2007889.2 |
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 siblings: | 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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(271\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |