Normalized defining polynomial
\( x^{18} - 3 x^{17} - 50 x^{16} + 169 x^{15} + 874 x^{14} - 3458 x^{13} - 5979 x^{12} + 31684 x^{11} + 8426 x^{10} - 131047 x^{9} + 53306 x^{8} + 228127 x^{7} - 156694 x^{6} - 157898 x^{5} + 123595 x^{4} + 40177 x^{3} - 33502 x^{2} - 1654 x + 2053 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(113290500653811459555808941573877=13^{15}\cdot 19^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $60.37$ | 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: | $13, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(247=13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{247}(64,·)$, $\chi_{247}(1,·)$, $\chi_{247}(68,·)$, $\chi_{247}(134,·)$, $\chi_{247}(140,·)$, $\chi_{247}(77,·)$, $\chi_{247}(144,·)$, $\chi_{247}(87,·)$, $\chi_{247}(153,·)$, $\chi_{247}(220,·)$, $\chi_{247}(30,·)$, $\chi_{247}(159,·)$, $\chi_{247}(235,·)$, $\chi_{247}(172,·)$, $\chi_{247}(49,·)$, $\chi_{247}(178,·)$, $\chi_{247}(121,·)$, $\chi_{247}(191,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{78231879476215077155680335392725977} a^{17} - \frac{124469085971059496177917595280718}{759532810448690069472624615463359} a^{16} + \frac{19859850204899748234178685678698784}{78231879476215077155680335392725977} a^{15} - \frac{2153004674168976526168709969752231}{78231879476215077155680335392725977} a^{14} + \frac{29148475360184763653507334814969235}{78231879476215077155680335392725977} a^{13} - \frac{20687767619851077348124354360147954}{78231879476215077155680335392725977} a^{12} + \frac{35965012878465387833704384045869442}{78231879476215077155680335392725977} a^{11} + \frac{1078565600509325303020338987298747}{26077293158738359051893445130908659} a^{10} + \frac{22347994212379905386281430551840628}{78231879476215077155680335392725977} a^{9} + \frac{10388222246609690766408959097925559}{26077293158738359051893445130908659} a^{8} - \frac{26647133077323386235515656641208777}{78231879476215077155680335392725977} a^{7} + \frac{24221905531217614628191387790675339}{78231879476215077155680335392725977} a^{6} - \frac{11992425851255249159281368539822594}{26077293158738359051893445130908659} a^{5} - \frac{16931014231891006496333894635294862}{78231879476215077155680335392725977} a^{4} + \frac{4016080529207884964615516327726353}{8692431052912786350631148376969553} a^{3} + \frac{2097842527036537377343058666997622}{78231879476215077155680335392725977} a^{2} - \frac{34672213633141542738179139092090738}{78231879476215077155680335392725977} a + \frac{23607734424142083397670540766947164}{78231879476215077155680335392725977}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 12940318365.3 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times C_6$ (as 18T2):
| An abelian group of order 18 |
| The 18 conjugacy class representatives for $C_6 \times C_3$ |
| Character table for $C_6 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 3.3.361.1, 3.3.61009.2, 3.3.61009.1, 3.3.169.1, 6.6.286315237.1, 6.6.48387275053.2, 6.6.48387275053.1, \(\Q(\zeta_{13})^+\), 9.9.227081481823729.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| 13 | Data not computed | ||||||
| 19 | Data not computed | ||||||