Normalized defining polynomial
\( x^{16} + 976 x^{14} + 346724 x^{12} + 55785232 x^{10} + 4316728379 x^{8} + 153444603544 x^{6} + 2044575845776 x^{4} + 8014083537528 x^{2} + 202716958081 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(182405472588468433947695572320256000000000000=2^{48}\cdot 5^{12}\cdot 61^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $583.87$ | 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: | $2, 5, 61$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4880=2^{4}\cdot 5\cdot 61\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4880}(1,·)$, $\chi_{4880}(4747,·)$, $\chi_{4880}(3783,·)$, $\chi_{4880}(843,·)$, $\chi_{4880}(1109,·)$, $\chi_{4880}(4381,·)$, $\chi_{4880}(3427,·)$, $\chi_{4880}(3049,·)$, $\chi_{4880}(2927,·)$, $\chi_{4880}(2929,·)$, $\chi_{4880}(4403,·)$, $\chi_{4880}(3061,·)$, $\chi_{4880}(2807,·)$, $\chi_{4880}(121,·)$, $\chi_{4880}(2429,·)$, $\chi_{4880}(3903,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{61} a^{4}$, $\frac{1}{61} a^{5}$, $\frac{1}{61} a^{6}$, $\frac{1}{61} a^{7}$, $\frac{1}{3721} a^{8}$, $\frac{1}{40931} a^{9} - \frac{1}{671} a^{7} - \frac{5}{671} a^{5} + \frac{5}{11} a^{3} + \frac{3}{11} a$, $\frac{1}{1186999} a^{10} + \frac{27}{1186999} a^{8} + \frac{94}{19459} a^{6} - \frac{14}{19459} a^{4} - \frac{30}{319} a^{2} - \frac{13}{29}$, $\frac{1}{1186999} a^{11} - \frac{2}{1186999} a^{9} + \frac{123}{19459} a^{7} + \frac{131}{19459} a^{5} + \frac{144}{319} a^{3} + \frac{89}{319} a$, $\frac{1}{28166299271} a^{12} - \frac{13}{461742611} a^{10} - \frac{16334}{461742611} a^{8} + \frac{1011}{7569551} a^{6} - \frac{26141}{7569551} a^{4} + \frac{13208}{124091} a^{2} + \frac{2229}{11281}$, $\frac{1}{309829291981} a^{13} + \frac{1932}{5079168721} a^{11} - \frac{42786}{5079168721} a^{9} + \frac{635081}{83265061} a^{7} - \frac{403082}{83265061} a^{5} - \frac{191795}{1365001} a^{3} - \frac{366426}{1365001} a$, $\frac{1}{97988227731829064881471} a^{14} + \frac{851338288931}{97988227731829064881471} a^{12} - \frac{98270069960443}{1606364389046378112811} a^{10} - \frac{70194160829050865}{1606364389046378112811} a^{8} - \frac{126461748215044647}{26333842443383247751} a^{6} + \frac{146294167348893757}{26333842443383247751} a^{4} - \frac{177700703431881382}{431702335137430291} a^{2} - \frac{522397058440039}{3567787893697771}$, $\frac{1}{97988227731829064881471} a^{15} - \frac{805432702}{809820063899413759351} a^{13} - \frac{578044401556890}{1606364389046378112811} a^{11} + \frac{6940078793486416}{1606364389046378112811} a^{9} - \frac{170111408168647303}{26333842443383247751} a^{7} + \frac{78088094754021406}{26333842443383247751} a^{5} - \frac{36325408218575017}{431702335137430291} a^{3} + \frac{90931612822498152}{431702335137430291} a$
Class group and class number
$C_{2}\times C_{2}\times C_{20}\times C_{16642600}$, which has order $1331408000$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 8592043.71726574 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4^2$ |
| Character table for $C_4^2$ |
Intermediate fields
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 | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $61$ | 61.8.6.2 | $x^{8} + 183 x^{4} + 14884$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 61.8.6.2 | $x^{8} + 183 x^{4} + 14884$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |