Normalized defining polynomial
\( x^{18} - 3 x^{17} - 14 x^{16} + 47 x^{15} + 79 x^{14} - 321 x^{13} - 234 x^{12} + 1356 x^{11} + 76 x^{10} - 3435 x^{9} + 1672 x^{8} + 4094 x^{7} - 3619 x^{6} - 1450 x^{5} + 2070 x^{4} - 70 x^{3} - 260 x^{2} + 11 x + 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(6576752670755330244294481=19^{16}\cdot 151^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.92$ | 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: | $19, 151$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{1}{7} a^{14} + \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{259} a^{16} + \frac{3}{259} a^{15} - \frac{116}{259} a^{14} + \frac{64}{259} a^{13} + \frac{101}{259} a^{12} + \frac{116}{259} a^{11} + \frac{34}{259} a^{10} + \frac{5}{37} a^{9} + \frac{128}{259} a^{8} - \frac{59}{259} a^{7} - \frac{93}{259} a^{6} - \frac{40}{259} a^{5} + \frac{86}{259} a^{4} + \frac{18}{259} a^{3} + \frac{72}{259} a^{2} + \frac{18}{37} a + \frac{107}{259}$, $\frac{1}{238810362847} a^{17} + \frac{4866123}{6454334131} a^{16} - \frac{5785272839}{238810362847} a^{15} - \frac{97760373763}{238810362847} a^{14} + \frac{68862794146}{238810362847} a^{13} - \frac{41998134642}{238810362847} a^{12} - \frac{71799231212}{238810362847} a^{11} - \frac{64198116392}{238810362847} a^{10} + \frac{14052586308}{34115766121} a^{9} + \frac{26899204315}{238810362847} a^{8} - \frac{16708219601}{238810362847} a^{7} + \frac{103699778065}{238810362847} a^{6} + \frac{16307666531}{34115766121} a^{5} + \frac{94222563398}{238810362847} a^{4} - \frac{43095888510}{238810362847} a^{3} - \frac{11670129815}{34115766121} a^{2} + \frac{23068560854}{238810362847} a + \frac{50010406434}{238810362847}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 1263648.42707 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 576 |
| The 16 conjugacy class representatives for t18n177 |
| Character table for t18n177 |
Intermediate fields
| 3.3.361.1, \(\Q(\zeta_{19})^+\) |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $19$ | 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| $151$ | $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 151.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.1.2 | $x^{2} + 755$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.1.2 | $x^{2} + 755$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |