Normalized defining polynomial
\( x^{18} - 2 x^{17} - 16 x^{16} - 5 x^{15} + 129 x^{14} + 198 x^{13} - 515 x^{12} - 850 x^{11} + 1039 x^{10} + 1049 x^{9} - 1411 x^{8} - 129 x^{7} + 1189 x^{6} - 284 x^{5} - 500 x^{4} + 26 x^{3} + 76 x^{2} + 6 x - 2 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1743338764216058776821760000=-\,2^{16}\cdot 5^{4}\cdot 37^{8}\cdot 59^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.61$ | 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: | $2, 5, 37, 59$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1344751547720369113} a^{17} - \frac{13850938072129185}{103442426747720701} a^{16} + \frac{12725471374544419}{1344751547720369113} a^{15} + \frac{237961175631905429}{1344751547720369113} a^{14} - \frac{328609994629034778}{1344751547720369113} a^{13} - \frac{503194123141330638}{1344751547720369113} a^{12} - \frac{23326242556176095}{1344751547720369113} a^{11} + \frac{612076927659885508}{1344751547720369113} a^{10} - \frac{53058917469527538}{1344751547720369113} a^{9} + \frac{212963454748374943}{1344751547720369113} a^{8} + \frac{536345314881427152}{1344751547720369113} a^{7} + \frac{385736667208318759}{1344751547720369113} a^{6} + \frac{503440816580290494}{1344751547720369113} a^{5} + \frac{436314527277453637}{1344751547720369113} a^{4} - \frac{292302722879060496}{1344751547720369113} a^{3} - \frac{25947155398646271}{1344751547720369113} a^{2} - \frac{318536528862182289}{1344751547720369113} a + \frac{556381608874937626}{1344751547720369113}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 38372119.4728 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 82944 |
| The 144 conjugacy class representatives for t18n772 are not computed |
| Character table for t18n772 is not computed |
Intermediate fields
| 3.3.148.1, 9.9.10438327105600.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 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 | R | $18$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.8.4 | $x^{6} + 2 x^{3} + 2 x^{2} + 2$ | $6$ | $1$ | $8$ | $S_4\times C_2$ | $[4/3, 4/3, 2]_{3}^{2}$ |
| 2.12.8.1 | $x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
| $5$ | 5.6.4.1 | $x^{6} + 25 x^{3} + 200$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.12.0.1 | $x^{12} - x^{3} - 2 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $37$ | $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.12.6.1 | $x^{12} + 2026120 x^{6} - 69343957 x^{2} + 1026290563600$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| $59$ | 59.6.0.1 | $x^{6} - x + 23$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 59.6.4.1 | $x^{6} + 295 x^{3} + 27848$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 59.6.0.1 | $x^{6} - x + 23$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |