Normalized defining polynomial
\( x^{16} + 100 x^{14} + 1694 x^{12} - 12716 x^{10} - 127655 x^{8} + 1043504 x^{6} - 1639792 x^{4} - 1288408 x^{2} + 1771561 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(374786530566485896134032948504855498981376=2^{48}\cdot 3^{12}\cdot 11^{8}\cdot 43^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $396.61$ | 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: | $2, 3, 11, 43$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} + \frac{1}{11} a^{4}$, $\frac{1}{11} a^{7} + \frac{1}{11} a^{5}$, $\frac{1}{121} a^{8} + \frac{1}{121} a^{6} + \frac{2}{11} a^{4} - \frac{1}{11} a^{2}$, $\frac{1}{121} a^{9} + \frac{1}{121} a^{7} + \frac{2}{11} a^{5} - \frac{1}{11} a^{3}$, $\frac{1}{1331} a^{10} + \frac{1}{1331} a^{8} + \frac{2}{121} a^{6} - \frac{45}{121} a^{4} - \frac{1}{11} a^{2}$, $\frac{1}{1331} a^{11} + \frac{1}{1331} a^{9} + \frac{2}{121} a^{7} - \frac{45}{121} a^{5} - \frac{1}{11} a^{3}$, $\frac{1}{3411353} a^{12} + \frac{958}{3411353} a^{10} + \frac{122}{310123} a^{8} + \frac{12913}{310123} a^{6} - \frac{431}{2563} a^{4} + \frac{430}{2563} a^{2} + \frac{6}{233}$, $\frac{1}{37524883} a^{13} + \frac{958}{37524883} a^{11} + \frac{7811}{3411353} a^{9} + \frac{133374}{3411353} a^{7} + \frac{7025}{28193} a^{5} + \frac{12546}{28193} a^{3} - \frac{227}{2563} a$, $\frac{1}{15272627381} a^{14} + \frac{47}{412773713} a^{12} + \frac{357759}{1388420671} a^{10} + \frac{3724588}{1388420671} a^{8} + \frac{553262}{126220061} a^{6} - \frac{5352671}{11474551} a^{4} + \frac{108824}{1043141} a^{2} - \frac{39184}{94831}$, $\frac{1}{15272627381} a^{15} + \frac{3}{412773713} a^{13} + \frac{215975}{1388420671} a^{11} + \frac{2482831}{1388420671} a^{9} + \frac{436923}{11474551} a^{7} - \frac{1142256}{11474551} a^{5} + \frac{243467}{1043141} a^{3} - \frac{508}{8621} a$
Class group and class number
$C_{2}\times C_{8}$, which has order $16$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 74326256631300 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1024 |
| The 55 conjugacy class representatives for t16n1220 are not computed |
| Character table for t16n1220 is not computed |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{86}) \), \(\Q(\sqrt{258}) \), 4.4.8786448.1, 4.4.76032.1, \(\Q(\sqrt{3}, \sqrt{86})\), 8.8.19763627124916224.1 |
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 | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| $11$ | 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.3.2 | $x^{4} - 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| $43$ | 43.8.4.1 | $x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 43.8.4.1 | $x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |