Normalized defining polynomial
\( x^{14} - x^{13} - 2 x^{12} + 4 x^{11} - 2 x^{10} - 3 x^{9} + x^{8} + 6 x^{6} - 11 x^{5} + 5 x^{4} + \cdots + 5 \)
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: | \(386441371953125\) \(\medspace = 5^{7}\cdot 53^{2}\cdot 1327^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.01\) | 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: | $5^{1/2}53^{1/2}1327^{1/2}\approx 593.0050590003427$ | ||
Ramified primes: | \(5\), \(53\), \(1327\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{27}a^{12}+\frac{1}{3}a^{10}+\frac{13}{27}a^{9}+\frac{2}{27}a^{8}+\frac{7}{27}a^{7}+\frac{1}{9}a^{6}-\frac{1}{27}a^{5}+\frac{11}{27}a^{4}-\frac{11}{27}a^{3}+\frac{7}{27}a^{2}+\frac{1}{3}a+\frac{2}{27}$, $\frac{1}{28593}a^{13}-\frac{182}{28593}a^{12}+\frac{130}{3177}a^{11}-\frac{11615}{28593}a^{10}-\frac{4523}{9531}a^{9}+\frac{4291}{9531}a^{8}+\frac{8260}{28593}a^{7}-\frac{1870}{28593}a^{6}-\frac{14171}{28593}a^{5}-\frac{1751}{9531}a^{4}+\frac{875}{28593}a^{3}-\frac{9068}{28593}a^{2}-\frac{10708}{28593}a-\frac{6196}{28593}$
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: | $\frac{1991}{28593}a^{13}+\frac{16760}{28593}a^{12}+\frac{434}{3177}a^{11}-\frac{31852}{28593}a^{10}+\frac{1855}{9531}a^{9}-\frac{328}{3177}a^{8}-\frac{48265}{28593}a^{7}-\frac{60089}{28593}a^{6}-\frac{38707}{28593}a^{5}+\frac{967}{353}a^{4}-\frac{7343}{28593}a^{3}-\frac{17500}{28593}a^{2}-\frac{189401}{28593}a+\frac{97483}{28593}$, $\frac{8035}{9531}a^{13}+\frac{169}{1059}a^{12}-\frac{2095}{1059}a^{11}+\frac{7381}{9531}a^{10}+\frac{1724}{9531}a^{9}-\frac{22685}{9531}a^{8}-\frac{6419}{3177}a^{7}-\frac{13066}{9531}a^{6}+\frac{51233}{9531}a^{5}-\frac{12353}{9531}a^{4}-\frac{1841}{9531}a^{3}-\frac{10087}{1059}a^{2}+\frac{83336}{9531}a-\frac{1004}{1059}$, $\frac{24157}{28593}a^{13}+\frac{394}{28593}a^{12}-\frac{5879}{3177}a^{11}+\frac{37678}{28593}a^{10}-\frac{845}{9531}a^{9}-\frac{24767}{9531}a^{8}-\frac{48674}{28593}a^{7}-\frac{34774}{28593}a^{6}+\frac{126337}{28593}a^{5}-\frac{36335}{9531}a^{4}+\frac{10325}{28593}a^{3}-\frac{277925}{28593}a^{2}+\frac{284114}{28593}a-\frac{109966}{28593}$, $\frac{3437}{9531}a^{13}-\frac{7078}{9531}a^{12}-\frac{500}{353}a^{11}+\frac{17312}{9531}a^{10}+\frac{24}{353}a^{9}-\frac{3350}{3177}a^{8}+\frac{11597}{9531}a^{7}+\frac{25297}{9531}a^{6}+\frac{52751}{9531}a^{5}-\frac{12245}{3177}a^{4}+\frac{4051}{9531}a^{3}-\frac{36352}{9531}a^{2}+\frac{126052}{9531}a-\frac{30932}{9531}$, $\frac{4184}{9531}a^{13}-\frac{1832}{9531}a^{12}-\frac{1112}{1059}a^{11}+\frac{10940}{9531}a^{10}-\frac{1471}{9531}a^{9}-\frac{14228}{9531}a^{8}-\frac{3449}{9531}a^{7}-\frac{1247}{9531}a^{6}+\frac{8650}{3177}a^{5}-\frac{27953}{9531}a^{4}+\frac{2248}{3177}a^{3}-\frac{45962}{9531}a^{2}+\frac{76030}{9531}a-\frac{27631}{9531}$, $\frac{26185}{28593}a^{13}+\frac{889}{28593}a^{12}-\frac{6989}{3177}a^{11}+\frac{33559}{28593}a^{10}+\frac{680}{3177}a^{9}-\frac{25834}{9531}a^{8}-\frac{58187}{28593}a^{7}-\frac{21088}{28593}a^{6}+\frac{173287}{28593}a^{5}-\frac{24181}{9531}a^{4}-\frac{2767}{28593}a^{3}-\frac{297356}{28593}a^{2}+\frac{327563}{28593}a-\frac{60646}{28593}$, $\frac{596}{28593}a^{13}-\frac{454}{28593}a^{12}+\frac{173}{3177}a^{11}+\frac{6497}{28593}a^{10}-\frac{553}{9531}a^{9}-\frac{1108}{9531}a^{8}-\frac{1390}{28593}a^{7}-\frac{8924}{28593}a^{6}-\frac{14158}{28593}a^{5}-\frac{12130}{9531}a^{4}-\frac{18590}{28593}a^{3}-\frac{16336}{28593}a^{2}+\frac{13333}{28593}a-\frac{36089}{28593}$ | 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: | \( 38.0258403894 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 38.0258403894 \cdot 1}{2\cdot\sqrt{386441371953125}}\cr\approx \mathstrut & 0.238037959694 \end{aligned}\]
Galois group
$C_{7236}$ (as 14T49):
A non-solvable group of order 10080 |
The 30 conjugacy class representatives for $S_7\times C_2$ |
Character table for $S_7\times C_2$ is not computed |
Intermediate fields
\(\Q(\sqrt{5}) \), 7.1.351655.1 |
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 sibling: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
\(53\) | 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
53.10.0.1 | $x^{10} + x^{6} + x^{4} + 27 x^{3} + 15 x^{2} + 29 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(1327\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |