Normalized defining polynomial
\( x^{16} - 12 x^{14} + 22 x^{12} - 408 x^{10} + 5971 x^{8} + 10320 x^{6} + 116250 x^{4} - 148500 x^{2} + 50625 \)
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: | \(35103376218247434118103040000=2^{36}\cdot 3^{10}\cdot 5^{4}\cdot 7^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $60.83$ | 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: | $2, 3, 5, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{20} a^{10} - \frac{1}{10} a^{8} - \frac{3}{20} a^{6} + \frac{1}{10} a^{4} - \frac{9}{20} a^{2} - \frac{1}{2}$, $\frac{1}{20} a^{11} - \frac{1}{10} a^{9} - \frac{3}{20} a^{7} + \frac{1}{10} a^{5} - \frac{9}{20} a^{3} - \frac{1}{2} a$, $\frac{1}{11100} a^{12} + \frac{43}{1850} a^{10} - \frac{1343}{11100} a^{8} - \frac{103}{1850} a^{6} - \frac{3989}{11100} a^{4} - \frac{87}{370} a^{2} - \frac{2}{37}$, $\frac{1}{11100} a^{13} + \frac{43}{1850} a^{11} - \frac{1343}{11100} a^{9} - \frac{103}{1850} a^{7} - \frac{3989}{11100} a^{5} - \frac{87}{370} a^{3} - \frac{2}{37} a$, $\frac{1}{31233871197000} a^{14} - \frac{1}{22200} a^{13} - \frac{193922527}{5205645199500} a^{12} + \frac{99}{7400} a^{11} + \frac{378185126197}{31233871197000} a^{10} - \frac{1271}{11100} a^{9} - \frac{1167111976361}{10411290399000} a^{8} + \frac{3351}{7400} a^{7} - \frac{394144904137}{1837286541000} a^{6} + \frac{581}{5550} a^{5} + \frac{524177324003}{2082258079800} a^{4} - \frac{159}{1480} a^{3} - \frac{1939317844}{10411290399} a^{2} - \frac{29}{296} a - \frac{12270398995}{27763441064}$, $\frac{1}{93701613591000} a^{15} - \frac{856822099}{31233871197000} a^{13} - \frac{1}{22200} a^{12} + \frac{199010918323}{23425403397750} a^{11} + \frac{99}{7400} a^{10} + \frac{2846393574749}{31233871197000} a^{9} - \frac{1271}{11100} a^{8} + \frac{1137567250789}{2755929811500} a^{7} + \frac{3351}{7400} a^{6} - \frac{2381229265181}{6246774239400} a^{5} + \frac{581}{5550} a^{4} - \frac{122313123853}{1249354847880} a^{3} - \frac{159}{1480} a^{2} + \frac{2221224645}{6940860266} a - \frac{29}{296}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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: | \( 7147945.41424 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^3$ (as 16T329):
| A solvable group of order 128 |
| The 29 conjugacy class representatives for $C_2^4.C_2^3$ |
| Character table for $C_2^4.C_2^3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{3}) \), 4.0.49392.1 x2, 4.0.65856.1 x2, \(\Q(\sqrt{3}, \sqrt{7})\), 8.0.39033114624.4 |
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 | R | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | ||||||
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7 | Data not computed | ||||||