Normalized defining polynomial
\( x^{16} - 26 x^{14} - 8 x^{13} - 45 x^{12} - 184 x^{11} + 3026 x^{10} + 5228 x^{9} - 3491 x^{8} - 19536 x^{7} - 68646 x^{6} - 16032 x^{5} + 189433 x^{4} + 80328 x^{3} - 117522 x^{2} - 15876 x + 9558 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(20434372100444980042526777925435392=2^{31}\cdot 7^{2}\cdot 17^{8}\cdot 2297^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $139.44$ | 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: | $2, 7, 17, 2297$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{80} a^{12} - \frac{1}{40} a^{10} - \frac{3}{20} a^{5} - \frac{11}{80} a^{4} - \frac{3}{10} a^{3} - \frac{3}{40} a^{2} - \frac{3}{20} a + \frac{17}{40}$, $\frac{1}{6240} a^{13} - \frac{1}{416} a^{12} - \frac{151}{3120} a^{11} + \frac{19}{624} a^{10} - \frac{1}{26} a^{9} - \frac{29}{312} a^{8} - \frac{71}{312} a^{7} + \frac{347}{1560} a^{6} - \frac{227}{480} a^{5} + \frac{467}{2080} a^{4} - \frac{401}{1040} a^{3} - \frac{287}{1040} a^{2} + \frac{1067}{3120} a + \frac{27}{208}$, $\frac{1}{3407040} a^{14} + \frac{3}{94640} a^{13} + \frac{1441}{3407040} a^{12} - \frac{11737}{425880} a^{11} + \frac{11503}{189280} a^{10} + \frac{7379}{85176} a^{9} - \frac{6641}{170352} a^{8} - \frac{84463}{851760} a^{7} - \frac{92921}{486720} a^{6} - \frac{9331}{20280} a^{5} + \frac{27623}{1135680} a^{4} - \frac{4123}{40560} a^{3} + \frac{4405}{21294} a^{2} + \frac{2623}{283920} a + \frac{85719}{189280}$, $\frac{1}{52891301206811546256960} a^{15} - \frac{181787992612703}{5876811245201282917440} a^{14} - \frac{1612815295340330657}{52891301206811546256960} a^{13} + \frac{774732793033702385}{1511180034480329893056} a^{12} + \frac{22649537418771768185}{587681124520128291744} a^{11} - \frac{2360622827288087561}{290611545092371133280} a^{10} - \frac{60235304079716101043}{2644565060340577312848} a^{9} + \frac{49136959438192882429}{1652853162712860820530} a^{8} + \frac{4372473313030209928813}{52891301206811546256960} a^{7} + \frac{260178654393176669713}{2518633390800549821760} a^{6} + \frac{1411842844577021382865}{3526086747120769750464} a^{5} - \frac{1619450444055003801127}{17630433735603848752320} a^{4} - \frac{3241794093242201137027}{6611412650851443282120} a^{3} + \frac{1002362621659028891}{3245661586083182760} a^{2} + \frac{787470779134244840711}{2938405622600641458720} a + \frac{112201890787087071731}{979468540866880486240}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 2116146488420 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 4096 |
| The 94 conjugacy class representatives for t16n1581 are not computed |
| Character table for t16n1581 is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.9248.1, 4.4.663833.1, 4.4.21242656.2, 8.8.451250433934336.1 |
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 | R | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
| 2.4.9.3 | $x^{4} + 6 x^{2} + 10$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ | |
| 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.8.6 | $x^{4} + 6 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $D_{4}$ | $[2, 3]^{2}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 2297 | Data not computed | ||||||