Normalized defining polynomial
\( x^{20} - 5 x^{19} + 6 x^{18} + 14 x^{17} - 46 x^{16} + 25 x^{15} + 73 x^{14} - 122 x^{13} - 8 x^{12} + 193 x^{11} - 146 x^{10} - 144 x^{9} + 308 x^{8} - 127 x^{7} - 128 x^{6} + 161 x^{5} - 46 x^{4} - 26 x^{3} + 25 x^{2} - 8 x + 1 \)
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: | \(379379679498607236597\)\(\medspace = 3^{15}\cdot 31^{9}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $10.69$ | 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: | $3, 31$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $10$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6978527} a^{19} - \frac{502850}{6978527} a^{18} + \frac{2639465}{6978527} a^{17} - \frac{2706308}{6978527} a^{16} - \frac{1189948}{6978527} a^{15} - \frac{438476}{6978527} a^{14} - \frac{1096272}{6978527} a^{13} + \frac{110407}{6978527} a^{12} - \frac{3425638}{6978527} a^{11} - \frac{707323}{6978527} a^{10} - \frac{751820}{6978527} a^{9} + \frac{1184585}{6978527} a^{8} + \frac{3485122}{6978527} a^{7} - \frac{557869}{6978527} a^{6} - \frac{1191169}{6978527} a^{5} - \frac{574971}{6978527} a^{4} + \frac{918839}{6978527} a^{3} + \frac{718635}{6978527} a^{2} + \frac{68564}{6978527} a - \frac{3141208}{6978527}$
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: | \( \frac{294846}{18413} a^{19} - \frac{1356362}{18413} a^{18} + \frac{1211607}{18413} a^{17} + \frac{4680751}{18413} a^{16} - \frac{11744765}{18413} a^{15} + \frac{2419699}{18413} a^{14} + \frac{23070963}{18413} a^{13} - \frac{26770987}{18413} a^{12} - \frac{14299518}{18413} a^{11} + \frac{52373757}{18413} a^{10} - \frac{21096924}{18413} a^{9} - \frac{53479978}{18413} a^{8} + \frac{69996438}{18413} a^{7} - \frac{6459642}{18413} a^{6} - \frac{43411115}{18413} a^{5} + \frac{29757726}{18413} a^{4} + \frac{526458}{18413} a^{3} - \frac{8417968}{18413} a^{2} + \frac{3705327}{18413} a - \frac{536855}{18413} \) (order $6$) | 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: | \( 181.753135766 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_5\times C_5:D_4$ (as 20T53):
A solvable group of order 200 |
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed |
Character table for $C_5\times C_5:D_4$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.837.1, 10.0.224415603.1 |
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 | $20$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | $20$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ | $20$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | $20$ |
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 |
---|---|---|---|---|---|---|---|
3 | Data not computed | ||||||
$31$ | 31.10.9.3 | $x^{10} - 74431$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
31.10.0.1 | $x^{10} - x + 11$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |