Normalized defining polynomial
\( x^{18} - 3 x^{17} + 2 x^{16} - 12 x^{15} - 7 x^{14} + 155 x^{13} - 236 x^{12} + 184 x^{11} - 543 x^{10} + 514 x^{9} + 1549 x^{8} - 3935 x^{7} + 3767 x^{6} - 1897 x^{5} + 442 x^{4} + 267 x^{3} - 372 x^{2} + 150 x + 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(181792228022997449601438115909=61\cdot 1129^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.22$ | 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: | $61, 1129$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{93} a^{16} + \frac{38}{93} a^{15} - \frac{46}{93} a^{14} + \frac{35}{93} a^{13} - \frac{26}{93} a^{12} - \frac{19}{93} a^{11} + \frac{7}{93} a^{10} + \frac{16}{93} a^{9} + \frac{1}{3} a^{8} - \frac{10}{93} a^{7} - \frac{8}{93} a^{6} - \frac{14}{93} a^{5} + \frac{15}{31} a^{4} + \frac{19}{93} a^{3} + \frac{1}{31} a^{2} + \frac{8}{93} a - \frac{26}{93}$, $\frac{1}{220703467294635538986387} a^{17} + \frac{158727477385779028282}{220703467294635538986387} a^{16} - \frac{18929191675747417317913}{73567822431545179662129} a^{15} - \frac{10246785851262146728115}{73567822431545179662129} a^{14} - \frac{29839343126933306010586}{220703467294635538986387} a^{13} - \frac{14906089194096228293246}{220703467294635538986387} a^{12} + \frac{88308790885905339355430}{220703467294635538986387} a^{11} + \frac{11853945747969662579687}{73567822431545179662129} a^{10} + \frac{23227708754709942020953}{73567822431545179662129} a^{9} - \frac{10371295943256235903106}{220703467294635538986387} a^{8} - \frac{55400261039431483495870}{220703467294635538986387} a^{7} - \frac{11667696615901026875999}{73567822431545179662129} a^{6} + \frac{104292881323351946323217}{220703467294635538986387} a^{5} - \frac{3115648927727417687429}{220703467294635538986387} a^{4} - \frac{98809096205429194127515}{220703467294635538986387} a^{3} - \frac{88590525433908475656307}{220703467294635538986387} a^{2} - \frac{104535233056079973651250}{220703467294635538986387} a - \frac{6692851003848267552640}{220703467294635538986387}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 21196050.3435 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 9216 |
| The 88 conjugacy class representatives for t18n548 are not computed |
| Character table for t18n548 is not computed |
Intermediate fields
| 3.3.1129.1, 9.9.1624709678881.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | 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 | $18$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | $18$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $61$ | 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 1129 | Data not computed | ||||||