Normalized defining polynomial
\( x^{14} - 5 x^{13} + 11 x^{12} - 47 x^{11} + 204 x^{10} - 368 x^{9} - 44 x^{8} + 1008 x^{7} - 916 x^{6} + \cdots + 304 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1453458087070606032896\) \(\medspace = 2^{22}\cdot 809^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.48\) | 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: | $2^{7/3}809^{3/4}\approx 764.477511593336$ | ||
Ramified primes: | \(2\), \(809\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{809}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}$, $\frac{1}{5236090952}a^{13}+\frac{88848635}{2618045476}a^{12}+\frac{52029641}{1309022738}a^{11}-\frac{279168691}{5236090952}a^{10}+\frac{9226781}{2618045476}a^{9}+\frac{561468721}{5236090952}a^{8}-\frac{1160838147}{5236090952}a^{7}+\frac{483934559}{2618045476}a^{6}+\frac{460349545}{2618045476}a^{5}+\frac{1120148403}{2618045476}a^{4}-\frac{471055749}{2618045476}a^{3}-\frac{390924935}{1309022738}a^{2}-\frac{267204976}{654511369}a-\frac{113841689}{654511369}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
| |
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: | $\frac{210061233}{5236090952}a^{13}-\frac{163813507}{1309022738}a^{12}+\frac{993916701}{5236090952}a^{11}-\frac{7749706355}{5236090952}a^{10}+\frac{6964736439}{1309022738}a^{9}-\frac{21675606063}{5236090952}a^{8}-\frac{30550792619}{2618045476}a^{7}+\frac{26522550787}{1309022738}a^{6}+\frac{6485712587}{1309022738}a^{5}-\frac{57887022575}{2618045476}a^{4}-\frac{2199115505}{654511369}a^{3}+\frac{9893116355}{654511369}a^{2}+\frac{2542336962}{654511369}a-\frac{3369763573}{654511369}$, $\frac{98965461}{5236090952}a^{13}-\frac{284156745}{5236090952}a^{12}+\frac{362029663}{5236090952}a^{11}-\frac{3629468657}{5236090952}a^{10}+\frac{3076256089}{1309022738}a^{9}-\frac{3097022531}{2618045476}a^{8}-\frac{7231359577}{1309022738}a^{7}+\frac{4146984795}{654511369}a^{6}+\frac{6152250859}{1309022738}a^{5}-\frac{3414123567}{654511369}a^{4}-\frac{6385210377}{1309022738}a^{3}+\frac{2436214615}{1309022738}a^{2}+\frac{818621766}{654511369}a-\frac{435258793}{654511369}$, $\frac{29805837}{5236090952}a^{13}-\frac{45446501}{2618045476}a^{12}+\frac{132776343}{5236090952}a^{11}-\frac{1104442847}{5236090952}a^{10}+\frac{1961539753}{2618045476}a^{9}-\frac{2859576177}{5236090952}a^{8}-\frac{4254769751}{2618045476}a^{7}+\frac{1512404943}{654511369}a^{6}+\frac{1869704169}{1309022738}a^{5}-\frac{4562973569}{2618045476}a^{4}-\frac{2332214022}{654511369}a^{3}+\frac{1388848307}{654511369}a^{2}+\frac{2441305349}{654511369}a-\frac{1497491395}{654511369}$, $\frac{99376071}{5236090952}a^{13}-\frac{70256699}{1309022738}a^{12}+\frac{303211587}{5236090952}a^{11}-\frac{3273536245}{5236090952}a^{10}+\frac{5794743447}{2618045476}a^{9}-\frac{3334574967}{5236090952}a^{8}-\frac{5405585704}{654511369}a^{7}+\frac{27741297781}{2618045476}a^{6}+\frac{4695974480}{654511369}a^{5}-\frac{43748087413}{2618045476}a^{4}-\frac{2613318537}{1309022738}a^{3}+\frac{12825060681}{1309022738}a^{2}+\frac{2796768721}{654511369}a-\frac{2387250579}{654511369}$, $\frac{34431395}{654511369}a^{13}-\frac{817450959}{5236090952}a^{12}+\frac{136750689}{654511369}a^{11}-\frac{4936927105}{2618045476}a^{10}+\frac{17340449975}{2618045476}a^{9}-\frac{20007922905}{5236090952}a^{8}-\frac{44990039693}{2618045476}a^{7}+\frac{32197959489}{1309022738}a^{6}+\frac{7433827194}{654511369}a^{5}-\frac{76850260691}{2618045476}a^{4}-\frac{5958807785}{1309022738}a^{3}+\frac{20963039349}{1309022738}a^{2}+\frac{1500379293}{654511369}a-\frac{2993947289}{654511369}$, $\frac{261634117}{2618045476}a^{13}-\frac{202825208}{654511369}a^{12}+\frac{1210104749}{2618045476}a^{11}-\frac{9586913455}{2618045476}a^{10}+\frac{34433684531}{2618045476}a^{9}-\frac{6460992392}{654511369}a^{8}-\frac{78492630383}{2618045476}a^{7}+\frac{133019002933}{2618045476}a^{6}+\frac{18263979023}{1309022738}a^{5}-\frac{74826207573}{1309022738}a^{4}-\frac{10260963377}{1309022738}a^{3}+\frac{49570214725}{1309022738}a^{2}+\frac{5991693993}{654511369}a-\frac{7131338669}{654511369}$, $\frac{87979895}{1309022738}a^{13}-\frac{342287043}{1309022738}a^{12}+\frac{319701234}{654511369}a^{11}-\frac{14542572183}{5236090952}a^{10}+\frac{57587950547}{5236090952}a^{9}-\frac{75324067177}{5236090952}a^{8}-\frac{61740493093}{5236090952}a^{7}+\frac{111772391383}{2618045476}a^{6}-\frac{14817252435}{1309022738}a^{5}-\frac{101963108257}{2618045476}a^{4}+\frac{23113954227}{2618045476}a^{3}+\frac{41150186749}{1309022738}a^{2}-\frac{705489499}{654511369}a-\frac{9852740416}{654511369}$, $\frac{105468727}{654511369}a^{13}-\frac{3104369179}{5236090952}a^{12}+\frac{5387880795}{5236090952}a^{11}-\frac{32923796249}{5236090952}a^{10}+\frac{129197324953}{5236090952}a^{9}-\frac{18357659059}{654511369}a^{8}-\frac{106179152207}{2618045476}a^{7}+\frac{71295586998}{654511369}a^{6}-\frac{37347014903}{2618045476}a^{5}-\frac{146218061311}{1309022738}a^{4}+\frac{22990463447}{1309022738}a^{3}+\frac{116147924505}{1309022738}a^{2}-\frac{756353072}{654511369}a-\frac{27226355075}{654511369}$, $\frac{1949496609}{5236090952}a^{13}-\frac{8285186763}{5236090952}a^{12}+\frac{14984289635}{5236090952}a^{11}-\frac{79788210297}{5236090952}a^{10}+\frac{168544399305}{2618045476}a^{9}-\frac{228077348393}{2618045476}a^{8}-\frac{227293602551}{2618045476}a^{7}+\frac{815558412495}{2618045476}a^{6}-\frac{120497210107}{1309022738}a^{5}-\frac{412518278273}{1309022738}a^{4}+\frac{67629657750}{654511369}a^{3}+\frac{174151679440}{654511369}a^{2}-\frac{24362468661}{654511369}a-\frac{92144953831}{654511369}$ | 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: | \( 1157392.83214 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{4}\cdot 1157392.83214 \cdot 1}{2\cdot\sqrt{1453458087070606032896}}\cr\approx \mathstrut & 1.51408056155 \end{aligned}\]
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
7.7.670188544.2 |
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 | R | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.14.0.1}{14} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.8.18.65 | $x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $S_4\times C_2$ | $[2, 8/3, 8/3]_{3}^{2}$ | |
\(809\) | $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $4$ | $1$ | $3$ |