Normalized defining polynomial
\(x^{20} - 3 x^{19} + 9 x^{18} - 18 x^{17} + 34 x^{16} - 53 x^{15} + 77 x^{14} - 97 x^{13} + 114 x^{12} - 121 x^{11} + 116 x^{10} - 104 x^{9} + 79 x^{8} - 56 x^{7} + 33 x^{6} - 19 x^{5} + 12 x^{4} - 7 x^{3} + 5 x^{2} - 2 x + 1\)
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: | \(2392595214874267578125\)\(\medspace = 5^{15}\cdot 280001^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $11.72$ | 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: | $5, 280001$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{17} - \frac{1}{7} a^{15} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{45787} a^{19} + \frac{1453}{45787} a^{18} - \frac{3707}{45787} a^{17} + \frac{387}{1477} a^{16} + \frac{22819}{45787} a^{15} - \frac{3869}{45787} a^{14} - \frac{198}{6541} a^{13} + \frac{16138}{45787} a^{12} - \frac{1616}{6541} a^{11} - \frac{216}{1477} a^{10} - \frac{22793}{45787} a^{9} - \frac{4219}{45787} a^{8} + \frac{2691}{6541} a^{7} - \frac{6338}{45787} a^{6} - \frac{18367}{45787} a^{5} - \frac{15845}{45787} a^{4} + \frac{6340}{45787} a^{3} - \frac{11400}{45787} a^{2} - \frac{554}{6541} a + \frac{18136}{45787}$
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: | \( -\frac{19365}{45787} a^{19} + \frac{80529}{45787} a^{18} - \frac{197650}{45787} a^{17} + \frac{13336}{1477} a^{16} - \frac{699485}{45787} a^{15} + \frac{1068754}{45787} a^{14} - \frac{1430312}{45787} a^{13} + \frac{242521}{6541} a^{12} - \frac{1799985}{45787} a^{11} + \frac{56946}{1477} a^{10} - \frac{1543911}{45787} a^{9} + \frac{1089651}{45787} a^{8} - \frac{765314}{45787} a^{7} + \frac{248604}{45787} a^{6} - \frac{74953}{45787} a^{5} + \frac{65525}{45787} a^{4} - \frac{25694}{45787} a^{3} + \frac{93824}{45787} a^{2} + \frac{19872}{45787} a + \frac{15355}{45787} \) (order $10$) ![]() | 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: | \( 858.376938924 \) (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 t20n654 are not computed |
Character table for t20n654 is not computed |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.875003125.1 |
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 | $20$ | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
5 | Data not computed | ||||||
280001 | Data not computed |