Normalized defining polynomial
\( x^{16} - 3 x^{15} + 8 x^{14} - 19 x^{13} + 27 x^{12} - 64 x^{11} + 77 x^{10} - 120 x^{9} + 218 x^{8} - 112 x^{7} + 517 x^{6} - 168 x^{5} + 250 x^{4} - 355 x^{3} + 428 x^{2} - 215 x + 59 \)
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: | \(1681397159705859765625=5^{8}\cdot 31^{2}\cdot 41^{4}\cdot 1259^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $21.21$ | 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: | $5, 31, 41, 1259$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{139} a^{14} + \frac{41}{139} a^{13} + \frac{21}{139} a^{12} + \frac{32}{139} a^{11} - \frac{36}{139} a^{10} - \frac{24}{139} a^{9} - \frac{26}{139} a^{8} + \frac{20}{139} a^{7} - \frac{13}{139} a^{6} + \frac{53}{139} a^{5} - \frac{15}{139} a^{3} + \frac{7}{139} a^{2} - \frac{9}{139} a + \frac{5}{139}$, $\frac{1}{77489553852241793} a^{15} - \frac{81546514095115}{77489553852241793} a^{14} + \frac{12645682608196085}{77489553852241793} a^{13} - \frac{3506879953465864}{77489553852241793} a^{12} + \frac{31985071573310130}{77489553852241793} a^{11} - \frac{499730781644773}{77489553852241793} a^{10} - \frac{1019468537945043}{77489553852241793} a^{9} - \frac{354544183243493}{1091402166932983} a^{8} - \frac{2205666230159761}{77489553852241793} a^{7} + \frac{1418883574611766}{77489553852241793} a^{6} - \frac{1789642600067919}{77489553852241793} a^{5} + \frac{9144239772001407}{77489553852241793} a^{4} - \frac{30089741077165028}{77489553852241793} a^{3} + \frac{1671099605401276}{77489553852241793} a^{2} + \frac{5358886405760133}{77489553852241793} a + \frac{3732847262651939}{77489553852241793}$
Class group and class number
Trivial group, which has order $1$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 6371.32285993 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 4608 |
| The 65 conjugacy class representatives for t16n1648 are not computed |
| Character table for t16n1648 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.1025.1, 8.8.1000118125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $31$ | 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.6.0.1 | $x^{6} - 2 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $41$ | 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 1259 | Data not computed | ||||||