Normalized defining polynomial
\(x^{30} - x^{29} + 5 x^{28} - 10 x^{27} + 31 x^{26} - 76 x^{25} + 210 x^{24} - 545 x^{23} + 1461 x^{22} - 3851 x^{21} + 10240 x^{20} + 18326 x^{19} + 26485 x^{18} + 36579 x^{17} + 51035 x^{16} + 68796 x^{15} + 98765 x^{14} + 125384 x^{13} + 200880 x^{12} + 201891 x^{11} + 476245 x^{10} + 130439 x^{9} + 35726 x^{8} + 9785 x^{7} + 2680 x^{6} + 734 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1\) 
Invariants
| Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-249154964698353870876083085574129912384252301255171\)\(\medspace = -\,11^{27}\cdot 13^{20}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $47.85$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $11, 13$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $30$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(143=11\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{143}(1,·)$, $\chi_{143}(3,·)$, $\chi_{143}(68,·)$, $\chi_{143}(133,·)$, $\chi_{143}(9,·)$, $\chi_{143}(74,·)$, $\chi_{143}(139,·)$, $\chi_{143}(14,·)$, $\chi_{143}(79,·)$, $\chi_{143}(16,·)$, $\chi_{143}(81,·)$, $\chi_{143}(131,·)$, $\chi_{143}(87,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(29,·)$, $\chi_{143}(94,·)$, $\chi_{143}(35,·)$, $\chi_{143}(100,·)$, $\chi_{143}(40,·)$, $\chi_{143}(105,·)$, $\chi_{143}(42,·)$, $\chi_{143}(107,·)$, $\chi_{143}(48,·)$, $\chi_{143}(113,·)$, $\chi_{143}(53,·)$, $\chi_{143}(118,·)$, $\chi_{143}(120,·)$, $\chi_{143}(61,·)$, $\chi_{143}(126,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{5379069389} a^{21} - \frac{1152358415}{5379069389} a^{20} - \frac{2103974700}{5379069389} a^{19} - \frac{2505458961}{5379069389} a^{18} + \frac{620987965}{5379069389} a^{17} + \frac{2219289669}{5379069389} a^{16} - \frac{2608948237}{5379069389} a^{15} + \frac{106980170}{5379069389} a^{14} - \frac{2003924009}{5379069389} a^{13} - \frac{338276463}{5379069389} a^{12} - \frac{2405330354}{5379069389} a^{11} + \frac{1528096971}{5379069389} a^{10} + \frac{53078293}{5379069389} a^{9} - \frac{2458425821}{5379069389} a^{8} + \frac{1142642022}{5379069389} a^{7} - \frac{271284821}{5379069389} a^{6} + \frac{1921209341}{5379069389} a^{5} + \frac{1230078742}{5379069389} a^{4} + \frac{1346974054}{5379069389} a^{3} + \frac{1652131573}{5379069389} a^{2} + \frac{2505685901}{5379069389} a - \frac{2623203052}{5379069389}$, $\frac{1}{5379069389} a^{22} - \frac{1528051540}{5379069389} a^{11} - \frac{235676305}{5379069389}$, $\frac{1}{5379069389} a^{23} - \frac{1528051540}{5379069389} a^{12} - \frac{235676305}{5379069389} a$, $\frac{1}{5379069389} a^{24} - \frac{1528051540}{5379069389} a^{13} - \frac{235676305}{5379069389} a^{2}$, $\frac{1}{5379069389} a^{25} - \frac{1528051540}{5379069389} a^{14} - \frac{235676305}{5379069389} a^{3}$, $\frac{1}{5379069389} a^{26} - \frac{1528051540}{5379069389} a^{15} - \frac{235676305}{5379069389} a^{4}$, $\frac{1}{5379069389} a^{27} - \frac{1528051540}{5379069389} a^{16} - \frac{235676305}{5379069389} a^{5}$, $\frac{1}{5379069389} a^{28} - \frac{1528051540}{5379069389} a^{17} - \frac{235676305}{5379069389} a^{6}$, $\frac{1}{5379069389} a^{29} - \frac{1528051540}{5379069389} a^{18} - \frac{235676305}{5379069389} a^{7}$
Class group and class number
$C_{427}$, which has order $427$ (assuming GRH)
Unit group
| Rank: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( \frac{44811}{5379069389} a^{24} + \frac{2035763030}{5379069389} a^{13} - \frac{71705594805}{5379069389} a^{2} \) (order $22$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 85915831770.81862 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
| A cyclic group of order 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 3.3.169.1, \(\Q(\zeta_{11})^+\), 6.0.38014691.1, \(\Q(\zeta_{11})\), 15.15.432659002790862279847129.1 |
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 | $30$ | $15^{2}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{6}$ | $30$ | R | R | $30$ | $30$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ | $30$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{6}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ | $15^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| 13 | Data not computed | ||||||