Normalized defining polynomial
\( x^{14} - 2 x^{13} - 4 x^{12} + 6 x^{11} + 10 x^{10} - 3 x^{9} - 19 x^{8} - 6 x^{7} + 27 x^{6} - x^{5} + \cdots + 1 \)
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: | \(422859037223221\) \(\medspace = 71^{6}\cdot 3301\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.08\) | 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: | $71^{1/2}3301^{1/2}\approx 484.1187870760646$ | ||
Ramified primes: | \(71\), \(3301\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{3301}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{923}a^{13}-\frac{116}{923}a^{12}+\frac{298}{923}a^{11}+\frac{185}{923}a^{10}+\frac{149}{923}a^{9}-\frac{375}{923}a^{8}+\frac{21}{71}a^{7}+\frac{254}{923}a^{6}-\frac{316}{923}a^{5}+\frac{2}{71}a^{4}-\frac{215}{923}a^{3}-\frac{402}{923}a^{2}-\frac{318}{923}a+\frac{251}{923}$
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: | $\frac{359}{923}a^{13}+\frac{814}{923}a^{12}-\frac{3778}{923}a^{11}-\frac{4656}{923}a^{10}+\frac{8264}{923}a^{9}+\frac{15824}{923}a^{8}+\frac{13}{71}a^{7}-\frac{26035}{923}a^{6}-\frac{14683}{923}a^{5}+\frac{1712}{71}a^{4}+\frac{4962}{923}a^{3}-\frac{9560}{923}a^{2}+\frac{1213}{923}a+\frac{578}{923}$, $\frac{742}{923}a^{13}-\frac{1156}{923}a^{12}-\frac{3173}{923}a^{11}+\frac{2512}{923}a^{10}+\frac{7182}{923}a^{9}+\frac{2342}{923}a^{8}-\frac{748}{71}a^{7}-\frac{8131}{923}a^{6}+\frac{11046}{923}a^{5}-\frac{7}{71}a^{4}-\frac{9081}{923}a^{3}+\frac{4460}{923}a^{2}+\frac{1255}{923}a-\frac{1127}{923}$, $\frac{181}{923}a^{13}-\frac{690}{923}a^{12}-\frac{519}{923}a^{11}+\frac{3026}{923}a^{10}+\frac{2048}{923}a^{9}-\frac{5111}{923}a^{8}-\frac{601}{71}a^{7}+\frac{2593}{923}a^{6}+\frac{13875}{923}a^{5}-\frac{64}{71}a^{4}-\frac{9379}{923}a^{3}+\frac{3847}{923}a^{2}+\frac{2437}{923}a-\frac{1642}{923}$, $\frac{1127}{923}a^{13}-\frac{1512}{923}a^{12}-\frac{5664}{923}a^{11}+\frac{3589}{923}a^{10}+\frac{13782}{923}a^{9}+\frac{3801}{923}a^{8}-\frac{1467}{71}a^{7}-\frac{16486}{923}a^{6}+\frac{22298}{923}a^{5}+\frac{763}{71}a^{4}-\frac{22631}{923}a^{3}+\frac{1062}{923}a^{2}+\frac{8045}{923}a-\frac{3253}{923}$, $\frac{62}{71}a^{13}-\frac{92}{71}a^{12}-\frac{197}{71}a^{11}+\frac{39}{71}a^{10}+\frac{363}{71}a^{9}+\frac{606}{71}a^{8}-\frac{185}{71}a^{7}-\frac{795}{71}a^{6}-\frac{280}{71}a^{5}-\frac{376}{71}a^{4}+\frac{799}{71}a^{3}+\frac{139}{71}a^{2}-\frac{617}{71}a+\frac{226}{71}$, $\frac{64}{923}a^{13}-\frac{40}{923}a^{12}+\frac{612}{923}a^{11}-\frac{2005}{923}a^{10}-\frac{2463}{923}a^{9}+\frac{5536}{923}a^{8}+\frac{563}{71}a^{7}-\frac{2204}{923}a^{6}-\frac{14686}{923}a^{5}-\frac{369}{71}a^{4}+\frac{19468}{923}a^{3}-\frac{2653}{923}a^{2}-\frac{9276}{923}a+\frac{4065}{923}$, $\frac{1273}{923}a^{13}-\frac{2757}{923}a^{12}-\frac{4614}{923}a^{11}+\frac{8447}{923}a^{10}+\frac{11538}{923}a^{9}-\frac{6645}{923}a^{8}-\frac{1880}{71}a^{7}-\frac{1554}{923}a^{6}+\frac{38926}{923}a^{5}-\frac{507}{71}a^{4}-\frac{30946}{923}a^{3}+\frac{11595}{923}a^{2}+\frac{7767}{923}a-\frac{5373}{923}$ | 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: | \( 32.6924434913 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 32.6924434913 \cdot 1}{2\cdot\sqrt{422859037223221}}\cr\approx \mathstrut & 0.195640506822 \end{aligned}\]
Galois group
$C_2\wr D_7$ (as 14T38):
A solvable group of order 1792 |
The 40 conjugacy class representatives for $C_2\wr D_7$ |
Character table for $C_2\wr D_7$ |
Intermediate fields
7.1.357911.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 siblings: | data not computed |
Degree 28 siblings: | data not computed |
Degree 32 sibling: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{5}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(71\) | 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(3301\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |