Normalized defining polynomial
\( x^{14} - 6 x^{13} + 12 x^{12} - 8 x^{11} - 11 x^{10} + 61 x^{9} - 142 x^{8} + 239 x^{7} - 391 x^{6} + \cdots - 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9269035929372191597\) \(\medspace = 53^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.64\) | 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: | $53^{5/6}\approx 27.346173066268356$ | ||
Ramified primes: | \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{53}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{46358198}a^{13}-\frac{17590195}{46358198}a^{12}-\frac{17387659}{46358198}a^{11}+\frac{9967515}{46358198}a^{10}-\frac{6430154}{23179099}a^{9}-\frac{9417277}{46358198}a^{8}-\frac{20259991}{46358198}a^{7}-\frac{58865}{23179099}a^{6}-\frac{20470477}{46358198}a^{5}-\frac{1795295}{46358198}a^{4}+\frac{18067}{277594}a^{3}-\frac{9963703}{23179099}a^{2}-\frac{6985898}{23179099}a-\frac{17283405}{46358198}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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)
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Fundamental units: | $a$, $\frac{122110535}{46358198}a^{13}-\frac{680875151}{46358198}a^{12}+\frac{1178643915}{46358198}a^{11}-\frac{488681077}{46358198}a^{10}-\frac{765163052}{23179099}a^{9}+\frac{6793025461}{46358198}a^{8}-\frac{14487456251}{46358198}a^{7}+\frac{11576939417}{23179099}a^{6}-\frac{38173415507}{46358198}a^{5}+\frac{43538892247}{46358198}a^{4}-\frac{172447585}{277594}a^{3}+\frac{6657014713}{23179099}a^{2}-\frac{1676437485}{23179099}a+\frac{338443065}{46358198}$, $\frac{62368599}{46358198}a^{13}-\frac{347275581}{46358198}a^{12}+\frac{599231611}{46358198}a^{11}-\frac{244044655}{46358198}a^{10}-\frac{395703610}{23179099}a^{9}+\frac{3479061617}{46358198}a^{8}-\frac{7368162171}{46358198}a^{7}+\frac{5874222522}{23179099}a^{6}-\frac{19330996729}{46358198}a^{5}+\frac{21915353903}{46358198}a^{4}-\frac{86027123}{277594}a^{3}+\frac{3142512975}{23179099}a^{2}-\frac{643771210}{23179099}a+\frac{110718297}{46358198}$, $\frac{47143377}{46358198}a^{13}-\frac{278297151}{46358198}a^{12}+\frac{528187019}{46358198}a^{11}-\frac{278391759}{46358198}a^{10}-\frac{298229871}{23179099}a^{9}+\frac{2805009891}{46358198}a^{8}-\frac{6301261013}{46358198}a^{7}+\frac{5098963351}{23179099}a^{6}-\frac{16687708489}{46358198}a^{5}+\frac{20105251713}{46358198}a^{4}-\frac{84168171}{277594}a^{3}+\frac{3317852305}{23179099}a^{2}-\frac{959224144}{23179099}a+\frac{193153431}{46358198}$, $\frac{3382775}{46358198}a^{13}-\frac{4191651}{46358198}a^{12}-\frac{44640691}{46358198}a^{11}+\frac{106044587}{46358198}a^{10}-\frac{34113770}{23179099}a^{9}-\frac{9985639}{46358198}a^{8}+\frac{366211601}{46358198}a^{7}-\frac{435813747}{23179099}a^{6}+\frac{1282159993}{46358198}a^{5}-\frac{2658448317}{46358198}a^{4}+\frac{19544495}{277594}a^{3}-\frac{909753796}{23179099}a^{2}+\frac{320666609}{23179099}a-\frac{69986225}{46358198}$, $\frac{94368325}{46358198}a^{13}-\frac{527446151}{46358198}a^{12}+\frac{917634965}{46358198}a^{11}-\frac{387619383}{46358198}a^{10}-\frac{590720405}{23179099}a^{9}+\frac{5264151089}{46358198}a^{8}-\frac{11262459643}{46358198}a^{7}+\frac{9012189231}{23179099}a^{6}-\frac{29688106743}{46358198}a^{5}+\frac{34000043931}{46358198}a^{4}-\frac{135305945}{277594}a^{3}+\frac{5224489494}{23179099}a^{2}-\frac{1350686755}{23179099}a+\frac{268120589}{46358198}$, $\frac{16742709}{23179099}a^{13}-\frac{96242787}{23179099}a^{12}+\frac{173749121}{23179099}a^{11}-\frac{74524234}{23179099}a^{10}-\frac{227257810}{23179099}a^{9}+\frac{966684485}{23179099}a^{8}-\frac{2089591610}{23179099}a^{7}+\frac{3322880448}{23179099}a^{6}-\frac{5406195579}{23179099}a^{5}+\frac{6325424696}{23179099}a^{4}-\frac{24339050}{138797}a^{3}+\frac{1585267829}{23179099}a^{2}-\frac{439557642}{23179099}a+\frac{50187517}{23179099}$, $\frac{161823485}{46358198}a^{13}-\frac{880899425}{46358198}a^{12}+\frac{1457815227}{46358198}a^{11}-\frac{504497397}{46358198}a^{10}-\frac{1025758479}{23179099}a^{9}+\frac{8744426125}{46358198}a^{8}-\frac{18175623359}{46358198}a^{7}+\frac{14383659192}{23179099}a^{6}-\frac{47505162299}{46358198}a^{5}+\frac{52662426509}{46358198}a^{4}-\frac{198554443}{277594}a^{3}+\frac{7348094735}{23179099}a^{2}-\frac{1612127387}{23179099}a+\frac{276332911}{46358198}$, $\frac{15256237}{23179099}a^{13}-\frac{62423192}{23179099}a^{12}+\frac{31359843}{23179099}a^{11}+\frac{106718763}{23179099}a^{10}-\frac{214643605}{23179099}a^{9}+\frac{561136575}{23179099}a^{8}-\frac{674282549}{23179099}a^{7}+\frac{703793371}{23179099}a^{6}-\frac{1384760340}{23179099}a^{5}-\frac{211358050}{23179099}a^{4}+\frac{12461655}{138797}a^{3}-\frac{1441097667}{23179099}a^{2}+\frac{791162426}{23179099}a-\frac{110576834}{23179099}$ | 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: | \( 25636.9775584 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{4}\cdot 25636.9775584 \cdot 1}{2\cdot\sqrt{9269035929372191597}}\cr\approx \mathstrut & 0.419970700039 \end{aligned}\]
Galois group
$\PGL(2,7)$ (as 14T16):
A non-solvable group of order 336 |
The 9 conjugacy class representatives for $\PGL(2,7)$ |
Character table for $\PGL(2,7)$ |
Intermediate fields
\(\Q(\sqrt{53}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | 8.0.418195493.1 |
Degree 16 sibling: | data not computed |
Degree 21 sibling: | data not computed |
Degree 24 sibling: | data not computed |
Degree 28 siblings: | data not computed |
Degree 42 siblings: | data not computed |
Minimal sibling: | 8.0.418195493.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/59.7.0.1}{7} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(53\) | 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
53.6.5.1 | $x^{6} + 53$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
53.6.5.1 | $x^{6} + 53$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |