Normalized defining polynomial
\( x^{16} - 72 x^{14} + 2442 x^{12} - 49008 x^{10} + 624187 x^{8} - 5024112 x^{6} + 24057244 x^{4} - 58068192 x^{2} + 54434884 \)
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: | \(204626334874637321888403030016=2^{56}\cdot 7^{6}\cdot 17^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $67.91$ | 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: | $2, 7, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{34} a^{12} - \frac{2}{17} a^{10} - \frac{3}{17} a^{8} - \frac{7}{17} a^{6} + \frac{15}{34} a^{4}$, $\frac{1}{68} a^{13} + \frac{15}{34} a^{11} + \frac{7}{17} a^{9} + \frac{5}{17} a^{7} + \frac{15}{68} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{136327255418497921484} a^{14} - \frac{54048925607788920}{34081813854624480371} a^{12} - \frac{6558159790915094161}{34081813854624480371} a^{10} + \frac{10124861231722402212}{34081813854624480371} a^{8} + \frac{39190286969434416435}{136327255418497921484} a^{6} - \frac{4697707536310083264}{34081813854624480371} a^{4} + \frac{1453475172119534045}{4009625159367585926} a^{2} + \frac{110803116251468685}{286401797097684709}$, $\frac{1}{4226144917973435566004} a^{15} + \frac{9807867195987809135}{4226144917973435566004} a^{13} + \frac{750717273277694930581}{2113072458986717783002} a^{11} + \frac{421111440066899959627}{1056536229493358891501} a^{9} - \frac{2077891797176650952493}{4226144917973435566004} a^{7} + \frac{1631169922934521275493}{4226144917973435566004} a^{5} - \frac{16314169341252473163}{62149189970197581853} a^{3} - \frac{2928813535571594429}{17756911420056451958} a$
Class group and class number
$C_{4}$, which has order $4$ (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: | \( 163331999.533 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^2:D_4$ (as 16T225):
| A solvable group of order 128 |
| The 32 conjugacy class representatives for $C_2^2.C_2^2:D_4$ |
| Character table for $C_2^2.C_2^2:D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.7168.1, 4.4.121856.1, 4.0.1088.2, 8.0.14848884736.9 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.28.74 | $x^{8} + 12 x^{4} + 28$ | $8$ | $1$ | $28$ | $Z_8 : Z_8^\times$ | $[2, 3, 7/2, 9/2]^{2}$ |
| 2.8.28.74 | $x^{8} + 12 x^{4} + 28$ | $8$ | $1$ | $28$ | $Z_8 : Z_8^\times$ | $[2, 3, 7/2, 9/2]^{2}$ | |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.2.1.1 | $x^{2} - 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.1 | $x^{2} - 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.8.6.2 | $x^{8} + 85 x^{4} + 2601$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |