Normalized defining polynomial
\( x^{14} - x^{13} + 4 x^{12} - x^{11} + 3 x^{10} + 3 x^{9} - 4 x^{8} + 9 x^{7} - 16 x^{6} + 14 x^{5} + \cdots - 1 \)
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: | \(245928712415581\) \(\medspace = 13^{4}\cdot 61\cdot 109^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.66\) | 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: | $13^{1/2}61^{1/2}109^{1/2}\approx 294.0017006753532$ | ||
Ramified primes: | \(13\), \(61\), \(109\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{9209}a^{13}+\frac{314}{9209}a^{12}-\frac{2385}{9209}a^{11}+\frac{3862}{9209}a^{10}+\frac{945}{9209}a^{9}+\frac{2990}{9209}a^{8}+\frac{2528}{9209}a^{7}+\frac{4355}{9209}a^{6}-\frac{332}{9209}a^{5}-\frac{3267}{9209}a^{4}+\frac{2282}{9209}a^{3}+\frac{538}{9209}a^{2}+\frac{3699}{9209}a-\frac{4356}{9209}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{699}{9209}a^{13}-\frac{10739}{9209}a^{12}+\frac{8923}{9209}a^{11}-\frac{35535}{9209}a^{10}-\frac{2493}{9209}a^{9}-\frac{18851}{9209}a^{8}-\frac{37892}{9209}a^{7}+\frac{42011}{9209}a^{6}-\frac{75515}{9209}a^{5}+\frac{138334}{9209}a^{4}-\frac{90129}{9209}a^{3}+\frac{155046}{9209}a^{2}-\frac{29755}{9209}a+\frac{30962}{9209}$, $\frac{16444}{9209}a^{13}-\frac{12042}{9209}a^{12}+\frac{57445}{9209}a^{11}+\frac{1464}{9209}a^{10}+\frac{31624}{9209}a^{9}+\frac{55963}{9209}a^{8}-\frac{63457}{9209}a^{7}+\frac{114944}{9209}a^{6}-\frac{210278}{9209}a^{5}+\frac{140893}{9209}a^{4}-\frac{240901}{9209}a^{3}+\frac{61486}{9209}a^{2}-\frac{54343}{9209}a-\frac{2462}{9209}$, $a$, $\frac{8465}{9209}a^{13}-\frac{12600}{9209}a^{12}+\frac{33939}{9209}a^{11}-\frac{18538}{9209}a^{10}+\frac{15222}{9209}a^{9}+\frac{22436}{9209}a^{8}-\frac{48241}{9209}a^{7}+\frac{84329}{9209}a^{6}-\frac{148979}{9209}a^{5}+\frac{146816}{9209}a^{4}-\frac{169114}{9209}a^{3}+\frac{106223}{9209}a^{2}-\frac{44610}{9209}a+\frac{8505}{9209}$, $\frac{16742}{9209}a^{13}-\frac{10560}{9209}a^{12}+\frac{55808}{9209}a^{11}+\frac{10424}{9209}a^{10}+\frac{27755}{9209}a^{9}+\frac{62919}{9209}a^{8}-\frac{56042}{9209}a^{7}+\frac{105056}{9209}a^{6}-\frac{198706}{9209}a^{5}+\frac{106645}{9209}a^{4}-\frac{214704}{9209}a^{3}+\frac{10003}{9209}a^{2}-\frac{38703}{9209}a-\frac{11290}{9209}$, $\frac{11296}{9209}a^{13}-\frac{7730}{9209}a^{12}+\frac{41410}{9209}a^{11}+\frac{2119}{9209}a^{10}+\frac{29116}{9209}a^{9}+\frac{42473}{9209}a^{8}-\frac{37657}{9209}a^{7}+\frac{82483}{9209}a^{6}-\frac{149553}{9209}a^{5}+\frac{97730}{9209}a^{4}-\frac{191908}{9209}a^{3}+\frac{45353}{9209}a^{2}-\frac{61792}{9209}a-\frac{1689}{9209}$, $\frac{10631}{9209}a^{13}-\frac{4733}{9209}a^{12}+\frac{34278}{9209}a^{11}+\frac{12409}{9209}a^{10}+\frac{17694}{9209}a^{9}+\frac{43267}{9209}a^{8}-\frac{33530}{9209}a^{7}+\frac{59616}{9209}a^{6}-\frac{112953}{9209}a^{5}+\frac{41707}{9209}a^{4}-\frac{116281}{9209}a^{3}-\frac{17729}{9209}a^{2}-\frac{7570}{9209}a-\frac{14993}{9209}$ | 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: | \( 26.4300628611 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 26.4300628611 \cdot 1}{2\cdot\sqrt{245928712415581}}\cr\approx \mathstrut & 0.207397149876 \end{aligned}\]
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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(61\) | 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |