Normalized defining polynomial
\( x^{10} - 28x^{8} + 556x^{6} - 1528x^{4} + 18352x^{2} + 14048 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-534284452043128832\) \(\medspace = -\,2^{15}\cdot 439^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(59.26\) | 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: | $2^{3/2}439^{1/2}\approx 59.262129560116215$ | ||
Ramified primes: | \(2\), \(439\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-878}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{12}a^{4}-\frac{1}{3}$, $\frac{1}{12}a^{5}-\frac{1}{3}a$, $\frac{1}{24}a^{6}-\frac{1}{6}a^{2}$, $\frac{1}{24}a^{7}-\frac{1}{6}a^{3}$, $\frac{1}{7751952}a^{8}-\frac{19}{8282}a^{6}-\frac{7424}{484497}a^{4}+\frac{637}{8282}a^{2}+\frac{155587}{484497}$, $\frac{1}{7751952}a^{9}-\frac{19}{8282}a^{7}-\frac{7424}{484497}a^{5}+\frac{637}{8282}a^{3}+\frac{155587}{484497}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{20}$, which has order $20$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{655}{7751952}a^{8}-\frac{11}{4141}a^{6}+\frac{90499}{1937988}a^{4}-\frac{503}{4141}a^{2}+\frac{3616}{484497}$, $\frac{241}{1937988}a^{8}-\frac{319}{99384}a^{6}+\frac{59939}{968994}a^{4}-\frac{2341}{12423}a^{2}+\frac{1083790}{484497}$, $\frac{953}{3875976}a^{9}-\frac{2833}{1937988}a^{8}+\frac{79}{33128}a^{7}-\frac{12}{4141}a^{6}+\frac{85573}{1937988}a^{5}-\frac{12211}{484497}a^{4}+\frac{809}{8282}a^{3}-\frac{721}{8282}a^{2}+\frac{521155}{484497}a+\frac{780194}{484497}$, $\frac{6130}{484497}a^{9}-\frac{189997}{7751952}a^{8}-\frac{4246}{12423}a^{7}+\frac{39497}{49692}a^{6}+\frac{12001711}{1937988}a^{5}-\frac{15008483}{968994}a^{4}+\frac{469}{12423}a^{3}+\frac{775744}{12423}a^{2}-\frac{2328536}{484497}a+\frac{25876559}{484497}$ | 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: | \( 5257.98433890173 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 5257.98433890173 \cdot 20}{2\cdot\sqrt{534284452043128832}}\cr\approx \mathstrut & 0.704421109262057 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{-878}) \), 5.1.192721.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.285459875695026432742900224149684224.2 |
Degree 10 sibling: | 10.2.1217048865701888.3 |
Minimal sibling: | 10.2.1217048865701888.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.2.0.1}{2} }^{5}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
\(439\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |