Normalized defining polynomial
\( x^{20} - 4 x^{19} - 42 x^{18} + 167 x^{17} + 590 x^{16} - 2576 x^{15} - 2936 x^{14} + 18115 x^{13} - 262 x^{12} - 56895 x^{11} + 34045 x^{10} + 71526 x^{9} - 65218 x^{8} - 32272 x^{7} + 41379 x^{6} + 787 x^{5} - 8208 x^{4} + 1108 x^{3} + 440 x^{2} - 113 x + 7 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[20, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(28202516462559027773277004254095077=13^{9}\cdot 277^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $52.79$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $13, 277$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{711633837560240033557229797} a^{19} - \frac{18198800965976411322392055}{101661976794320004793889971} a^{18} + \frac{49444385485559875485904685}{101661976794320004793889971} a^{17} + \frac{118166978235143203214502359}{711633837560240033557229797} a^{16} + \frac{98023082049470549481349439}{711633837560240033557229797} a^{15} - \frac{33421748439306953804759598}{711633837560240033557229797} a^{14} - \frac{4108501013393835821670689}{101661976794320004793889971} a^{13} + \frac{329295163045660417515011269}{711633837560240033557229797} a^{12} - \frac{1842520865924077705335381}{5350630357595789725994209} a^{11} - \frac{104189880726457995383515796}{711633837560240033557229797} a^{10} - \frac{239974265747986897454077610}{711633837560240033557229797} a^{9} - \frac{144704187928471633584852572}{711633837560240033557229797} a^{8} - \frac{278825841120892415857632584}{711633837560240033557229797} a^{7} + \frac{284766897787499205986793063}{711633837560240033557229797} a^{6} + \frac{21773768469630632882849752}{101661976794320004793889971} a^{5} + \frac{258926529168915423272425524}{711633837560240033557229797} a^{4} + \frac{32186183205007751677470196}{711633837560240033557229797} a^{3} - \frac{172006299422474213376389760}{711633837560240033557229797} a^{2} - \frac{349772068987914051780195130}{711633837560240033557229797} a - \frac{38941377815249785107124856}{101661976794320004793889971}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 84757731927.8 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 40 |
| The 13 conjugacy class representatives for $D_{20}$ |
| Character table for $D_{20}$ |
Intermediate fields
| \(\Q(\sqrt{277}) \), 4.4.997477.1, 5.5.12967201.1, 10.10.46577079591509077.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 20 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | $20$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ | $20$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 277 | Data not computed | ||||||