Normalized defining polynomial
\( x^{20} - 12250x^{14} + 245000x^{12} + 2143750x^{10} + 58953125x^{8} + 20633593750x^{4} + 1805439453125 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(114631697410565840810117120000000000\) \(\medspace = 2^{20}\cdot 5^{10}\cdot 7^{15}\cdot 11^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(56.62\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2\cdot 5^{1/2}7^{3/4}11^{9/10}\approx 166.5681431875978$ | ||
Ramified primes: | \(2\), \(5\), \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{77}) \) | ||
$\card{ \Aut(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{175}a^{4}$, $\frac{1}{175}a^{5}$, $\frac{1}{875}a^{6}$, $\frac{1}{875}a^{7}$, $\frac{1}{30625}a^{8}$, $\frac{1}{30625}a^{9}$, $\frac{1}{612500}a^{10}-\frac{1}{700}a^{4}-\frac{1}{10}a^{2}+\frac{1}{4}$, $\frac{1}{612500}a^{11}-\frac{1}{700}a^{5}-\frac{1}{10}a^{3}+\frac{1}{4}a$, $\frac{1}{21437500}a^{12}+\frac{1}{3500}a^{6}-\frac{1}{350}a^{4}-\frac{1}{20}a^{2}$, $\frac{1}{21437500}a^{13}+\frac{1}{3500}a^{7}-\frac{1}{350}a^{5}-\frac{1}{20}a^{3}$, $\frac{1}{107187500}a^{14}-\frac{1}{122500}a^{8}-\frac{1}{1750}a^{6}+\frac{1}{700}a^{4}$, $\frac{1}{107187500}a^{15}-\frac{1}{122500}a^{9}-\frac{1}{1750}a^{7}+\frac{1}{700}a^{5}$, $\frac{1}{3751562500}a^{16}-\frac{1}{61250}a^{8}-\frac{1}{3500}a^{6}+\frac{1}{700}a^{4}-\frac{1}{10}a^{2}-\frac{1}{4}$, $\frac{1}{3751562500}a^{17}-\frac{1}{61250}a^{9}-\frac{1}{3500}a^{7}+\frac{1}{700}a^{5}-\frac{1}{10}a^{3}-\frac{1}{4}a$, $\frac{1}{54\!\cdots\!00}a^{18}+\frac{128621}{10\!\cdots\!50}a^{16}+\frac{418311}{310489280937500}a^{14}+\frac{1136783}{62097856187500}a^{12}+\frac{70447}{126730318750}a^{10}+\frac{1758431}{177422446250}a^{8}+\frac{910536}{2534606375}a^{6}-\frac{1748983}{2027685100}a^{4}-\frac{77243}{2896693}a^{2}-\frac{2321853}{5793386}$, $\frac{1}{54\!\cdots\!00}a^{19}+\frac{128621}{10\!\cdots\!50}a^{17}+\frac{418311}{310489280937500}a^{15}+\frac{1136783}{62097856187500}a^{13}+\frac{70447}{126730318750}a^{11}+\frac{1758431}{177422446250}a^{9}+\frac{910536}{2534606375}a^{7}-\frac{1748983}{2027685100}a^{5}-\frac{77243}{2896693}a^{3}-\frac{2321853}{5793386}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
$C_{10}\wr C_2$ (as 20T53):
A solvable group of order 200 |
The 65 conjugacy class representatives for $C_{10}\wr C_2$ |
Character table for $C_{10}\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 4.0.1509200.2, 10.0.246071287.1 |
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 40 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $20$ | R | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{10}$ | $20$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.10.11 | $x^{10} + 10 x^{9} + 38 x^{8} + 64 x^{7} + 152 x^{6} + 688 x^{5} + 912 x^{4} - 1024 x^{3} - 1968 x^{2} + 32 x - 32$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
2.10.10.7 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
\(5\) | Deg $20$ | $2$ | $10$ | $10$ | |||
\(7\) | 7.20.15.1 | $x^{20} + 39 x^{16} + 16 x^{15} - 344 x^{12} - 7232 x^{11} + 96 x^{10} + 4344 x^{8} + 131648 x^{7} + 38272 x^{6} + 256 x^{5} + 59536 x^{4} - 212224 x^{3} + 84096 x^{2} - 8704 x + 9328$ | $4$ | $5$ | $15$ | 20T12 | $[\ ]_{4}^{10}$ |
\(11\) | 11.10.9.5 | $x^{10} + 88$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |