Normalized defining polynomial
\( x^{20} - 6 x^{19} - 68 x^{18} + 546 x^{17} + 790 x^{16} - 15528 x^{15} + 27509 x^{14} + 121115 x^{13} - 534817 x^{12} + 534768 x^{11} + 742976 x^{10} - 1898411 x^{9} + 659006 x^{8} + 1365776 x^{7} - 1231009 x^{6} - 10130 x^{5} + 355693 x^{4} - 131936 x^{3} + 14573 x^{2} - 486 x - 1 \)
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: | \(823070770673063523939149301768798828125=5^{13}\cdot 19^{10}\cdot 43^{8}\cdot 97^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $88.26$ | 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: | $5, 19, 43, 97$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{1}{19} a^{17} + \frac{9}{19} a^{16} - \frac{5}{19} a^{15} - \frac{2}{19} a^{14} - \frac{8}{19} a^{13} - \frac{9}{19} a^{12} - \frac{6}{19} a^{11} + \frac{7}{19} a^{10} + \frac{3}{19} a^{9} - \frac{2}{19} a^{8} - \frac{1}{19} a^{7} - \frac{4}{19} a^{6} - \frac{4}{19} a^{5} - \frac{3}{19} a^{4} - \frac{7}{19} a^{2} + \frac{8}{19} a + \frac{3}{19}$, $\frac{1}{2132891971563230561648130027679748150807111} a^{19} + \frac{4562756719508398848306472800350181495599}{2132891971563230561648130027679748150807111} a^{18} - \frac{17915413861060110268786297382904822913000}{2132891971563230561648130027679748150807111} a^{17} - \frac{356573279981429835869491760049074086592310}{2132891971563230561648130027679748150807111} a^{16} - \frac{883274757752298906938885382953923587581425}{2132891971563230561648130027679748150807111} a^{15} + \frac{680212011097803585095064818659250448203777}{2132891971563230561648130027679748150807111} a^{14} - \frac{12457953626634208571128711435121006852418}{304698853080461508806875718239964021543873} a^{13} + \frac{36248904693139761156784268013643429603748}{112257472187538450613059475141039376358269} a^{12} - \frac{724554380929395811503309921209242311875880}{2132891971563230561648130027679748150807111} a^{11} - \frac{20721347050067730782032956834519333783852}{2132891971563230561648130027679748150807111} a^{10} - \frac{372867334982410788967202602980541690147301}{2132891971563230561648130027679748150807111} a^{9} + \frac{82923923492969130479338532443216635741726}{304698853080461508806875718239964021543873} a^{8} - \frac{1056871503040732324591656184207068894571519}{2132891971563230561648130027679748150807111} a^{7} - \frac{597510417442903800454415817414048642113858}{2132891971563230561648130027679748150807111} a^{6} - \frac{840761409938123407777402516012245123659305}{2132891971563230561648130027679748150807111} a^{5} - \frac{1027536921212471868975967457766300885559655}{2132891971563230561648130027679748150807111} a^{4} - \frac{805320078840386508739884915928271568117534}{2132891971563230561648130027679748150807111} a^{3} + \frac{55612074302445575219506201042396239277038}{112257472187538450613059475141039376358269} a^{2} + \frac{668257575008766862640854812792392781779753}{2132891971563230561648130027679748150807111} a - \frac{249650089109831832242825378476298395389467}{2132891971563230561648130027679748150807111}$
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: | \( 19253819955500 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 100 conjugacy class representatives for t20n426 are not computed |
| Character table for t20n426 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.667489.1, 10.10.1392317391003125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | $20$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | $20$ | R | $20$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | R | $20$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.3.2 | $x^{4} - 19$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.3.2 | $x^{4} - 19$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| $43$ | 43.4.0.1 | $x^{4} - x + 20$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 43.8.4.1 | $x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 43.8.4.1 | $x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $97$ | 97.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 97.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.4.0.1 | $x^{4} - x + 23$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.4.0.1 | $x^{4} - x + 23$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.4.2.2 | $x^{4} - 97 x^{2} + 47045$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |