Normalized defining polynomial
\( x^{20} + 2 x^{18} + x^{14} + 7 x^{12} - 6 x^{10} + 7 x^{8} + x^{6} + 2 x^{2} + 1 \)
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: | \(960210593667613720576\)\(\medspace = 2^{20}\cdot 5501^{4}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $11.20$ | 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: | $2, 5501$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $4$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{15} + \frac{1}{4} a^{13} - \frac{1}{8} a^{12} + \frac{3}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{14} - \frac{1}{8} a^{13} + \frac{1}{4} a^{12} + \frac{3}{8} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{16} - \frac{1}{4} a^{15} + \frac{3}{16} a^{14} - \frac{1}{4} a^{13} - \frac{1}{16} a^{10} + \frac{1}{4} a^{9} - \frac{3}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} - \frac{3}{16} a^{2} + \frac{1}{4} a + \frac{3}{16}$, $\frac{1}{16} a^{19} - \frac{1}{16} a^{17} + \frac{3}{16} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{16} a^{11} - \frac{3}{16} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{16} a^{5} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2} + \frac{3}{16} a + \frac{1}{4}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( -1 \) (order $2$) | 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 103.153871476 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A non-solvable group of order 61440 |
The 126 conjugacy class representatives for t20n669 are not computed |
Character table for t20n669 is not computed |
Intermediate fields
5.1.5501.1, 10.2.1936704064.1, 10.0.30987265024.2, 10.0.484176016.1 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | 2.8.8.2 | $x^{8} + 2 x^{7} + 8 x^{2} + 48$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
2.12.12.11 | $x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$ | $2$ | $6$ | $12$ | $A_4 \times C_2$ | $[2, 2]^{6}$ | |
5501 | Data not computed |