Normalized defining polynomial
\( x^{20} + x^{18} - 4 x^{17} + 2 x^{16} - 4 x^{15} + 7 x^{14} - 6 x^{13} + 8 x^{12} - 6 x^{11} + 7 x^{10} - 6 x^{9} + 4 x^{8} - 2 x^{7} - 2 x^{5} - x^{4} + x^{2} + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(3020631315406201552896\)\(\medspace = 2^{20}\cdot 3^{10}\cdot 220873^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $11.86$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $2, 3, 220873$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $2$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{17} + \frac{1}{5} a^{16} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{1745} a^{19} + \frac{12}{349} a^{18} - \frac{238}{1745} a^{17} + \frac{723}{1745} a^{16} - \frac{592}{1745} a^{15} - \frac{624}{1745} a^{14} - \frac{18}{349} a^{13} - \frac{869}{1745} a^{12} + \frac{567}{1745} a^{11} - \frac{537}{1745} a^{10} - \frac{803}{1745} a^{9} - \frac{373}{1745} a^{8} - \frac{738}{1745} a^{7} + \frac{41}{1745} a^{6} - \frac{332}{1745} a^{5} + \frac{64}{349} a^{4} - \frac{694}{1745} a^{3} + \frac{589}{1745} a^{2} + \frac{158}{349} a - \frac{413}{1745}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( \frac{57}{349} a^{19} - \frac{70}{349} a^{18} + \frac{45}{349} a^{17} - \frac{320}{349} a^{16} + \frac{458}{349} a^{15} - \frac{319}{349} a^{14} + \frac{803}{349} a^{13} - \frac{1022}{349} a^{12} + \frac{909}{349} a^{11} - \frac{944}{349} a^{10} + \frac{995}{349} a^{9} - \frac{1019}{349} a^{8} + \frac{512}{349} a^{7} - \frac{106}{349} a^{6} + \frac{271}{349} a^{5} + \frac{92}{349} a^{4} - \frac{121}{349} a^{3} + \frac{418}{349} a^{2} + \frac{9}{349} a - \frac{158}{349} \) (order $12$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 1171.34610492 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A non-solvable group of order 57600 |
The 70 conjugacy class representatives for t20n656 are not computed |
Character table for t20n656 is not computed |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 10.0.226173952.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | Data not computed | ||||||
3 | Data not computed | ||||||
220873 | Data not computed |