Normalized defining polynomial
\( x^{20} - 7 x^{19} + 24 x^{18} - 52 x^{17} + 75 x^{16} - 58 x^{15} - 28 x^{14} + 140 x^{13} - 109 x^{12} - 274 x^{11} + 1033 x^{10} - 1874 x^{9} + 2367 x^{8} - 2282 x^{7} + 1739 x^{6} - 1057 x^{5} + 509 x^{4} - 190 x^{3} + 53 x^{2} - 10 x + 1 \)
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: | \(2825584168318748978373\)\(\medspace = 3^{10}\cdot 37^{5}\cdot 109^{2}\cdot 241^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $11.82$ | 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: | $3, 37, 109, 241$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{58974943} a^{19} + \frac{26100956}{58974943} a^{18} + \frac{4441811}{58974943} a^{17} + \frac{6717049}{58974943} a^{16} + \frac{29168546}{58974943} a^{15} + \frac{21041664}{58974943} a^{14} - \frac{1661676}{58974943} a^{13} + \frac{8787212}{58974943} a^{12} + \frac{21434130}{58974943} a^{11} + \frac{27183508}{58974943} a^{10} - \frac{7725163}{58974943} a^{9} + \frac{15959240}{58974943} a^{8} + \frac{10227602}{58974943} a^{7} + \frac{4964115}{58974943} a^{6} - \frac{3726288}{58974943} a^{5} + \frac{1530909}{58974943} a^{4} + \frac{21410941}{58974943} a^{3} - \frac{27182349}{58974943} a^{2} + \frac{4596607}{58974943} a + \frac{17015652}{58974943}$
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: | \( \frac{603522}{140083} a^{19} - \frac{4064809}{140083} a^{18} + \frac{13277554}{140083} a^{17} - \frac{27061215}{140083} a^{16} + \frac{35690210}{140083} a^{15} - \frac{21127454}{140083} a^{14} - \frac{27499899}{140083} a^{13} + \frac{78545176}{140083} a^{12} - \frac{37794123}{140083} a^{11} - \frac{188496067}{140083} a^{10} + \frac{573679398}{140083} a^{9} - \frac{936961751}{140083} a^{8} + \frac{1079862509}{140083} a^{7} - \frac{948908718}{140083} a^{6} + \frac{654085764}{140083} a^{5} - \frac{354326695}{140083} a^{4} + \frac{148436724}{140083} a^{3} - \frac{46655711}{140083} a^{2} + \frac{10373793}{140083} a - \frac{1206961}{140083} \) (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 (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 567.746569822 \) (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 115200 |
The 119 conjugacy class representatives for t20n781 are not computed |
Character table for t20n781 is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.333.1, 10.0.236184579.1 |
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 | $20$ | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | $20$ |
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 |
---|---|---|---|---|---|---|---|
$3$ | 3.10.5.2 | $x^{10} - 81 x^{2} + 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
3.10.5.2 | $x^{10} - 81 x^{2} + 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
37 | Data not computed | ||||||
$109$ | $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
109.4.2.1 | $x^{4} + 1199 x^{2} + 427716$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
109.4.0.1 | $x^{4} - x + 30$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
109.4.0.1 | $x^{4} - x + 30$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
241 | Data not computed |