Normalized defining polynomial
\( x^{18} - 6 x^{17} + 11 x^{16} - 6 x^{15} - 21 x^{14} + 182 x^{13} - 493 x^{12} + 233 x^{11} + 1189 x^{10} - 1823 x^{9} - 580 x^{8} + 2290 x^{7} + 202 x^{6} - 1152 x^{5} - 959 x^{4} + 503 x^{3} + 670 x^{2} - 152 x + 8 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(279992823820843547402730217=7^{12}\cdot 53^{6}\cdot 97^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.46$ | 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: | $7, 53, 97$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{187633576903816435204508} a^{17} + \frac{15793237033581913284319}{93816788451908217602254} a^{16} - \frac{22723056812060795178357}{187633576903816435204508} a^{15} - \frac{10209819898782779606113}{93816788451908217602254} a^{14} + \frac{47065773665712069252031}{187633576903816435204508} a^{13} + \frac{44315281771468328170979}{93816788451908217602254} a^{12} - \frac{10743479071605235931069}{187633576903816435204508} a^{11} - \frac{93035654811798297522111}{187633576903816435204508} a^{10} - \frac{16877349329856326490479}{187633576903816435204508} a^{9} + \frac{89210462877770890067545}{187633576903816435204508} a^{8} - \frac{17154473962084280934272}{46908394225954108801127} a^{7} + \frac{23802742613025390165355}{93816788451908217602254} a^{6} - \frac{34586286905664825148761}{93816788451908217602254} a^{5} + \frac{21672589203328486958276}{46908394225954108801127} a^{4} - \frac{62731039252086920886019}{187633576903816435204508} a^{3} + \frac{10498668008448932360683}{187633576903816435204508} a^{2} - \frac{15859259925160342278295}{93816788451908217602254} a + \frac{11663246436820310949780}{46908394225954108801127}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 4746478.85652 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 576 |
| The 28 conjugacy class representatives for t18n175 |
| Character table for t18n175 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 16 sibling: | data not computed |
| 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 | ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| $53$ | 53.3.0.1 | $x^{3} - x + 8$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 53.3.0.1 | $x^{3} - x + 8$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 53.6.3.1 | $x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 53.6.3.1 | $x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 97 | Data not computed | ||||||