Normalized defining polynomial
\( x^{16} - 7 x^{15} + 21 x^{14} - 40 x^{13} + 80 x^{12} - 154 x^{11} + 305 x^{10} - 370 x^{9} + 540 x^{8} - 581 x^{7} + 459 x^{6} - 222 x^{5} - 43 x^{4} + 106 x^{3} - 23 x^{2} - 12 x + 4 \)
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: | \(31216831471804594140625=5^{8}\cdot 89^{6}\cdot 401^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.46$ | 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: | $5, 89, 401$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{3}{8} a^{10} + \frac{3}{8} a^{9} - \frac{1}{2} a^{7} + \frac{3}{8} a^{5} - \frac{3}{8} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{330444205711072} a^{15} + \frac{376763116284}{10326381428471} a^{14} + \frac{19745482115381}{330444205711072} a^{13} - \frac{76766779696157}{330444205711072} a^{12} + \frac{141909030416045}{330444205711072} a^{11} - \frac{13240222430719}{330444205711072} a^{10} - \frac{5275207244475}{20652762856942} a^{9} - \frac{52546941853793}{165222102855536} a^{8} - \frac{9067801889013}{165222102855536} a^{7} + \frac{25838084803213}{330444205711072} a^{6} - \frac{69151410030857}{165222102855536} a^{5} - \frac{23748074715097}{82611051427768} a^{4} + \frac{28050539122025}{330444205711072} a^{3} + \frac{142616284914969}{330444205711072} a^{2} - \frac{1302612668775}{41305525713884} a + \frac{36588114275229}{82611051427768}$
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: | \( 111797.949516 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2048 |
| The 77 conjugacy class representatives for t16n1429 are not computed |
| Character table for t16n1429 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.2225.1, 8.0.440605625.1 |
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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 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 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $89$ | 89.4.0.1 | $x^{4} - x + 27$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 89.4.0.1 | $x^{4} - x + 27$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.8.6.2 | $x^{8} + 979 x^{4} + 285156$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 401 | Data not computed | ||||||