Normalized defining polynomial
\( x^{16} - 3 x^{15} + 19 x^{14} - 53 x^{13} + 180 x^{12} - 395 x^{11} + 1026 x^{10} - 1834 x^{9} + 3659 x^{8} - 5009 x^{7} + 7596 x^{6} - 6585 x^{5} + 9765 x^{4} - 5098 x^{3} + 15289 x^{2} - 4768 x + 12251 \)
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: | \(4693679410268375297265625=5^{8}\cdot 13^{4}\cdot 29^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.83$ | 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: | $5, 13, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1945} a^{14} + \frac{115}{389} a^{13} - \frac{419}{1945} a^{12} + \frac{156}{389} a^{11} + \frac{902}{1945} a^{10} + \frac{871}{1945} a^{9} - \frac{272}{1945} a^{8} - \frac{393}{1945} a^{7} + \frac{161}{1945} a^{6} + \frac{518}{1945} a^{5} - \frac{263}{1945} a^{4} + \frac{112}{1945} a^{3} + \frac{156}{389} a^{2} + \frac{116}{1945} a - \frac{128}{1945}$, $\frac{1}{466509251658034078721496055} a^{15} - \frac{86224737038176776119006}{466509251658034078721496055} a^{14} + \frac{4850255543174138134852812}{20283010941653655596586785} a^{13} + \frac{95951252943908171789682959}{466509251658034078721496055} a^{12} + \frac{160608944143546574849927377}{466509251658034078721496055} a^{11} + \frac{209829693945812263576705644}{466509251658034078721496055} a^{10} + \frac{98448880748904575978780607}{466509251658034078721496055} a^{9} - \frac{108752445587351565573847626}{466509251658034078721496055} a^{8} + \frac{182233222259180431996487439}{466509251658034078721496055} a^{7} + \frac{231598962451487627104013042}{466509251658034078721496055} a^{6} + \frac{34210052070043492466416594}{466509251658034078721496055} a^{5} - \frac{2716518207690046072269034}{93301850331606815744299211} a^{4} + \frac{59403922604745662196198203}{466509251658034078721496055} a^{3} - \frac{230911718906803656019992194}{466509251658034078721496055} a^{2} - \frac{192583369478448736260641479}{466509251658034078721496055} a - \frac{129668631752885976707037937}{466509251658034078721496055}$
Class group and class number
$C_{2}\times C_{34}$, which has order $68$ (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: | \( 3793.72993285 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 64 |
| The 16 conjugacy class representatives for $D_4.D_4$ |
| Character table for $D_4.D_4$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $29$ | 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 29.8.6.2 | $x^{8} + 145 x^{4} + 7569$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |