Normalized defining polynomial
\( x^{14} - 4 x^{13} + 5 x^{12} + 4 x^{11} - 14 x^{10} + 5 x^{9} + 13 x^{8} - 9 x^{7} - 9 x^{6} + 8 x^{5} + \cdots - 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(391066968923137\) \(\medspace = 13^{4}\cdot 97\cdot 109^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.02\) | 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: | $13^{1/2}97^{1/2}109^{1/2}\approx 370.7411495909242$ | ||
Ramified primes: | \(13\), \(97\), \(109\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{97}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{367}a^{13}-\frac{152}{367}a^{12}+\frac{114}{367}a^{11}+\frac{14}{367}a^{10}+\frac{116}{367}a^{9}+\frac{86}{367}a^{8}+\frac{130}{367}a^{7}-\frac{165}{367}a^{6}-\frac{178}{367}a^{5}-\frac{72}{367}a^{4}+\frac{17}{367}a^{3}+\frac{48}{367}a^{2}-\frac{131}{367}a-\frac{62}{367}$
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)
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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: | $a$, $\frac{185}{367}a^{13}-\frac{595}{367}a^{12}+\frac{538}{367}a^{11}+\frac{755}{367}a^{10}-\frac{1294}{367}a^{9}-\frac{238}{367}a^{8}+\frac{1296}{367}a^{7}-\frac{64}{367}a^{6}-\frac{1368}{367}a^{5}+\frac{626}{367}a^{4}+\frac{209}{367}a^{3}-\frac{662}{367}a^{2}-\frac{13}{367}a+\frac{274}{367}$, $\frac{38}{367}a^{13}-\frac{271}{367}a^{12}+\frac{662}{367}a^{11}-\frac{569}{367}a^{10}-\frac{363}{367}a^{9}+\frac{699}{367}a^{8}+\frac{169}{367}a^{7}-\frac{398}{367}a^{6}-\frac{525}{367}a^{5}+\frac{567}{367}a^{4}-\frac{88}{367}a^{3}-\frac{11}{367}a^{2}-\frac{207}{367}a-\frac{154}{367}$, $\frac{35}{367}a^{13}-\frac{182}{367}a^{12}+\frac{320}{367}a^{11}+\frac{123}{367}a^{10}-\frac{1078}{367}a^{9}+\frac{808}{367}a^{8}+\frac{1247}{367}a^{7}-\frac{1738}{367}a^{6}-\frac{725}{367}a^{5}+\frac{1517}{367}a^{4}+\frac{228}{367}a^{3}-\frac{889}{367}a^{2}-\frac{181}{367}a+\frac{32}{367}$, $\frac{32}{367}a^{13}-\frac{93}{367}a^{12}-\frac{22}{367}a^{11}+\frac{448}{367}a^{10}-\frac{692}{367}a^{9}+\frac{183}{367}a^{8}+\frac{490}{367}a^{7}-\frac{876}{367}a^{6}+\frac{176}{367}a^{5}+\frac{632}{367}a^{4}-\frac{190}{367}a^{3}-\frac{666}{367}a^{2}+\frac{212}{367}a+\frac{218}{367}$, $\frac{143}{367}a^{13}-\frac{817}{367}a^{12}+\frac{1622}{367}a^{11}-\frac{567}{367}a^{10}-\frac{2496}{367}a^{9}+\frac{2756}{367}a^{8}+\frac{974}{367}a^{7}-\frac{2309}{367}a^{6}-\frac{498}{367}a^{5}+\frac{1448}{367}a^{4}+\frac{229}{367}a^{3}-\frac{476}{367}a^{2}-\frac{16}{367}a-\frac{58}{367}$, $\frac{33}{367}a^{13}-\frac{245}{367}a^{12}+\frac{459}{367}a^{11}+\frac{95}{367}a^{10}-\frac{1310}{367}a^{9}+\frac{636}{367}a^{8}+\frac{1721}{367}a^{7}-\frac{1408}{367}a^{6}-\frac{1470}{367}a^{5}+\frac{1294}{367}a^{4}+\frac{1295}{367}a^{3}-\frac{618}{367}a^{2}-\frac{653}{367}a+\frac{156}{367}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 35.8148412318 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 35.8148412318 \cdot 1}{2\cdot\sqrt{391066968923137}}\cr\approx \mathstrut & 0.222867462546 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(97\) | 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
97.6.0.1 | $x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
97.6.0.1 | $x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(109\) | $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | 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 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.8.4.1 | $x^{8} + 28776 x^{7} + 310522274 x^{6} + 1489272514288 x^{5} + 2678510521605233 x^{4} + 178743008720712 x^{3} + 29612720181709536 x^{2} + 263388846138180416 x + 20054316486246464$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |