Normalized defining polynomial
\( x^{16} + 92 x^{14} + 3481 x^{12} + 69871 x^{10} + 797640 x^{8} + 5120439 x^{6} + 16818149 x^{4} + 21568097 x^{2} + 3829849 \)
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: | \(1438086201607169204454400000000=2^{16}\cdot 5^{8}\cdot 19^{6}\cdot 103^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $76.71$ | 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: | $2, 5, 19, 103$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{103} a^{10} + \frac{12}{103} a^{8} + \frac{49}{103} a^{6} + \frac{31}{103} a^{4}$, $\frac{1}{103} a^{11} + \frac{12}{103} a^{9} + \frac{49}{103} a^{7} + \frac{31}{103} a^{5}$, $\frac{1}{206} a^{12} - \frac{1}{206} a^{10} + \frac{99}{206} a^{8} - \frac{91}{206} a^{6} - \frac{47}{103} a^{4} - \frac{1}{2}$, $\frac{1}{206} a^{13} - \frac{1}{206} a^{11} + \frac{99}{206} a^{9} - \frac{91}{206} a^{7} - \frac{47}{103} a^{5} - \frac{1}{2} a$, $\frac{1}{14221599229449894} a^{14} + \frac{14567445127214}{7110799614724947} a^{12} + \frac{1114050414866}{2370266538241649} a^{10} + \frac{555056638393724}{7110799614724947} a^{8} - \frac{5852274908425081}{14221599229449894} a^{6} - \frac{3227766905707267}{7110799614724947} a^{4} + \frac{23648452874623}{138073778926698} a^{2} + \frac{2153625969733}{7267040996142}$, $\frac{1}{14221599229449894} a^{15} + \frac{14567445127214}{7110799614724947} a^{13} + \frac{1114050414866}{2370266538241649} a^{11} + \frac{555056638393724}{7110799614724947} a^{9} - \frac{5852274908425081}{14221599229449894} a^{7} - \frac{3227766905707267}{7110799614724947} a^{5} + \frac{23648452874623}{138073778926698} a^{3} + \frac{2153625969733}{7267040996142} a$
Class group and class number
$C_{2}\times C_{4}\times C_{2424}$, which has order $19392$ (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: | \( 10491.0986114 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 768 |
| The 31 conjugacy class representatives for t16n1048 |
| Character table for t16n1048 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.8 | $x^{8} + 4 x^{5} + 8 x^{2} + 48$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
| 2.8.8.7 | $x^{8} + 2 x^{6} + 4 x^{5} + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
| 5 | Data not computed | ||||||
| $19$ | 19.4.2.2 | $x^{4} - 19 x^{2} + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $103$ | 103.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 103.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 103.8.6.1 | $x^{8} - 3193 x^{4} + 6630625$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |