Normalized defining polynomial
\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 1026 x^{12} + 6702 x^{11} - 15504 x^{10} + 15370 x^{9} + 280960 x^{8} - 1141194 x^{7} + 1930786 x^{6} - 1808530 x^{5} - 13917442 x^{4} + 29524424 x^{3} - 20810458 x^{2} + 5935959 x - 493501 \)
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: | \(2356327674777993305467029827040553=61^{2}\cdot 97^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $121.83$ | 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: | $61, 97$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{39425017156339829983781249} a^{14} - \frac{7}{39425017156339829983781249} a^{13} - \frac{76881275797596252336047}{540068728169038766901113} a^{12} - \frac{5751018356992671460592572}{39425017156339829983781249} a^{11} + \frac{9754631884086167183532137}{39425017156339829983781249} a^{10} - \frac{2626327340721319192859150}{39425017156339829983781249} a^{9} + \frac{18211690996996113773508583}{39425017156339829983781249} a^{8} - \frac{2074950401556436052053691}{39425017156339829983781249} a^{7} - \frac{16281634525883597377875}{540068728169038766901113} a^{6} - \frac{7906399418905814506354143}{39425017156339829983781249} a^{5} - \frac{4167280783733513058848052}{39425017156339829983781249} a^{4} - \frac{13857664010912818077280533}{39425017156339829983781249} a^{3} + \frac{105295694764148361930680}{646311756661308688258709} a^{2} + \frac{8795172504741270342292253}{39425017156339829983781249} a + \frac{16292445003958247756799325}{39425017156339829983781249}$, $\frac{1}{4060776767103002488329468647} a^{15} + \frac{44}{4060776767103002488329468647} a^{14} + \frac{1531963335964028842946936923}{4060776767103002488329468647} a^{13} - \frac{94854922369744368988789308}{4060776767103002488329468647} a^{12} - \frac{1150897681762016336949876513}{4060776767103002488329468647} a^{11} + \frac{21759692871595247361904849}{4060776767103002488329468647} a^{10} - \frac{1377331552382665724543308035}{4060776767103002488329468647} a^{9} + \frac{138220947318448766721259062}{4060776767103002488329468647} a^{8} + \frac{642064296170689028428520615}{4060776767103002488329468647} a^{7} - \frac{1172423405136285687090057740}{4060776767103002488329468647} a^{6} + \frac{1051331983636643656516996868}{4060776767103002488329468647} a^{5} + \frac{1981411976773708495013218759}{4060776767103002488329468647} a^{4} + \frac{718982790452293207552589261}{4060776767103002488329468647} a^{3} + \frac{1716245679387900873740981448}{4060776767103002488329468647} a^{2} - \frac{11122610734231016955999213}{66570110936114794890647027} a - \frac{824936025364402223722046883}{4060776767103002488329468647}$
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: | \( 19911357834.4 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 32 conjugacy class representatives for t16n841 |
| Character table for t16n841 is not computed |
Intermediate fields
| \(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
| $61$ | $\Q_{61}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{61}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{61}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{61}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 97 | Data not computed | ||||||