Normalized defining polynomial
\( x^{10} - 4x^{9} + 5x^{8} + 15x^{7} + 12x^{6} - 79x^{5} + 142x^{4} - 103x^{3} + 619x^{2} - 128x + 443 \)
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)
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Discriminant: | \(-16305067506199\) \(\medspace = -\,439^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(20.95\) | 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: | $439^{1/2}\approx 20.952326839756964$ | ||
Ramified primes: | \(439\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-439}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{39}a^{6}-\frac{4}{39}a^{5}+\frac{4}{39}a^{4}+\frac{1}{39}a^{3}+\frac{4}{13}a^{2}-\frac{10}{39}a+\frac{3}{13}$, $\frac{1}{39}a^{7}+\frac{1}{39}a^{5}+\frac{4}{39}a^{4}-\frac{10}{39}a^{3}+\frac{4}{13}a^{2}+\frac{8}{39}a-\frac{16}{39}$, $\frac{1}{117}a^{8}+\frac{1}{117}a^{7}-\frac{1}{117}a^{6}-\frac{1}{9}a^{5}+\frac{4}{39}a^{4}-\frac{2}{9}a^{3}-\frac{10}{39}a^{2}+\frac{17}{39}a+\frac{31}{117}$, $\frac{1}{10027251}a^{9}+\frac{14291}{10027251}a^{8}+\frac{19867}{3342417}a^{7}+\frac{108490}{10027251}a^{6}+\frac{84401}{771327}a^{5}-\frac{2503067}{10027251}a^{4}+\frac{4630651}{10027251}a^{3}-\frac{927997}{3342417}a^{2}-\frac{3619469}{10027251}a-\frac{2008304}{10027251}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{444}{1114139}a^{9}+\frac{3182}{1114139}a^{8}-\frac{1491}{85703}a^{7}+\frac{7625}{257109}a^{6}+\frac{112423}{1114139}a^{5}-\frac{3919}{85703}a^{4}-\frac{1389026}{3342417}a^{3}+\frac{3178106}{3342417}a^{2}+\frac{2146429}{3342417}a+\frac{519796}{257109}$, $\frac{8794}{10027251}a^{9}-\frac{51247}{10027251}a^{8}+\frac{3668}{257109}a^{7}-\frac{70439}{10027251}a^{6}-\frac{152539}{10027251}a^{5}+\frac{158728}{10027251}a^{4}+\frac{2821237}{10027251}a^{3}-\frac{769511}{1114139}a^{2}+\frac{4083340}{10027251}a-\frac{312677}{771327}$, $\frac{9218}{3342417}a^{9}-\frac{11125}{771327}a^{8}+\frac{17159}{771327}a^{7}+\frac{406054}{10027251}a^{6}-\frac{218927}{10027251}a^{5}-\frac{388992}{1114139}a^{4}+\frac{5542097}{10027251}a^{3}-\frac{865451}{3342417}a^{2}+\frac{5483353}{3342417}a-\frac{14607454}{10027251}$, $\frac{4309}{3342417}a^{9}-\frac{35911}{10027251}a^{8}-\frac{7843}{10027251}a^{7}+\frac{349150}{10027251}a^{6}+\frac{168763}{10027251}a^{5}-\frac{54853}{1114139}a^{4}+\frac{650909}{10027251}a^{3}+\frac{77204}{1114139}a^{2}+\frac{351386}{1114139}a+\frac{16004819}{10027251}$ | 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: | \( 159.412146987 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 159.412146987 \cdot 3}{2\cdot\sqrt{16305067506199}}\cr\approx \mathstrut & 0.579896798645 \end{aligned}\]
Galois group
A solvable group of order 10 |
The 4 conjugacy class representatives for $D_5$ |
Character table for $D_5$ |
Intermediate fields
\(\Q(\sqrt{-439}) \), 5.1.192721.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.192721.1 |
Minimal sibling: | 5.1.192721.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{5}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.1.0.1}{1} }^{10}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(439\) | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |