Normalized defining polynomial
\( x^{16} + 9 x^{14} - 958 x^{12} - 2538 x^{10} + 249086 x^{8} - 421095 x^{6} - 11931762 x^{4} + 2406104 x^{2} + 20151121 \)
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: | \(222363776235307126475417780224=2^{16}\cdot 17^{14}\cdot 67^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $68.26$ | 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: | $2, 17, 67$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{134} a^{10} - \frac{29}{67} a^{8} + \frac{47}{134} a^{6} + \frac{4}{67} a^{4} + \frac{47}{134} a^{2} - \frac{1}{2}$, $\frac{1}{134} a^{11} - \frac{29}{67} a^{9} + \frac{47}{134} a^{7} + \frac{4}{67} a^{5} + \frac{47}{134} a^{3} - \frac{1}{2} a$, $\frac{1}{17956} a^{12} + \frac{9}{17956} a^{10} + \frac{3531}{17956} a^{8} + \frac{1951}{17956} a^{6} - \frac{6787}{17956} a^{4} + \frac{20}{67} a^{2} - \frac{1}{4}$, $\frac{1}{17956} a^{13} + \frac{9}{17956} a^{11} + \frac{3531}{17956} a^{9} + \frac{1951}{17956} a^{7} - \frac{6787}{17956} a^{5} + \frac{20}{67} a^{3} - \frac{1}{4} a$, $\frac{1}{87467627012348953754248} a^{14} - \frac{722423628738358797}{43733813506174476877124} a^{12} + \frac{64511985859329084281}{21866906753087238438562} a^{10} + \frac{11406549972943021363099}{43733813506174476877124} a^{8} + \frac{4182628692767303807105}{10933453376543619219281} a^{6} + \frac{590843981647437741791}{1305486970333566473944} a^{4} + \frac{8346110194910251035}{19484880154232335432} a^{2} - \frac{80415302200680251}{290819106779587096}$, $\frac{1}{87467627012348953754248} a^{15} - \frac{722423628738358797}{43733813506174476877124} a^{13} + \frac{64511985859329084281}{21866906753087238438562} a^{11} + \frac{11406549972943021363099}{43733813506174476877124} a^{9} + \frac{4182628692767303807105}{10933453376543619219281} a^{7} + \frac{590843981647437741791}{1305486970333566473944} a^{5} + \frac{8346110194910251035}{19484880154232335432} a^{3} - \frac{80415302200680251}{290819106779587096} a$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 251720438.635 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^3.C_2$ (as 16T565):
| A solvable group of order 256 |
| The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$ |
| Character table for $C_2^4.C_2^3.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.3 | $x^{8} + 2 x^{7} + 2 x^{6} + 16$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ |
| 2.8.8.3 | $x^{8} + 2 x^{7} + 2 x^{6} + 16$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ | |
| $17$ | 17.8.7.3 | $x^{8} - 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.3 | $x^{8} - 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
| $67$ | $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} - 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |