Normalized defining polynomial
\( x^{10} - x^{9} + 47x^{8} - 43x^{7} + 807x^{6} - 574x^{5} + 5961x^{4} - 2941x^{3} + 16861x^{2} - 6550x + 10991 \)
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: | \(-3618214860447423\) \(\medspace = -\,3^{8}\cdot 223^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.96\) | 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: | $3^{4/5}223^{1/2}\approx 35.962463598304126$ | ||
Ramified primes: | \(3\), \(223\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-223}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{389392534508767}a^{9}-\frac{42287998005670}{389392534508767}a^{8}-\frac{37234202812136}{389392534508767}a^{7}+\frac{132111653052941}{389392534508767}a^{6}-\frac{151285822455927}{389392534508767}a^{5}-\frac{66871243161563}{389392534508767}a^{4}-\frac{106280748805189}{389392534508767}a^{3}-\frac{141838818652287}{389392534508767}a^{2}+\frac{150617822840218}{389392534508767}a-\frac{73956428741897}{389392534508767}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{21}$, which has order $21$
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: | $\frac{36241637263}{389392534508767}a^{9}-\frac{154542153677}{389392534508767}a^{8}+\frac{1848178595444}{389392534508767}a^{7}-\frac{4751197838518}{389392534508767}a^{6}+\frac{31575307318229}{389392534508767}a^{5}-\frac{39724572363033}{389392534508767}a^{4}+\frac{169557153028447}{389392534508767}a^{3}-\frac{22813777147401}{389392534508767}a^{2}+\frac{83482510005112}{389392534508767}a+\frac{270878593742534}{389392534508767}$, $\frac{60847821114}{389392534508767}a^{9}-\frac{183541755783}{389392534508767}a^{8}+\frac{2052872007001}{389392534508767}a^{7}-\frac{6422940533520}{389392534508767}a^{6}+\frac{20858699765241}{389392534508767}a^{5}-\frac{54003508540646}{389392534508767}a^{4}+\frac{50347020393304}{389392534508767}a^{3}+\frac{23839720192430}{389392534508767}a^{2}-\frac{24017629674592}{389392534508767}a+\frac{904288890794128}{389392534508767}$, $\frac{107281785686}{389392534508767}a^{9}-\frac{457376820714}{389392534508767}a^{8}+\frac{5014962803597}{389392534508767}a^{7}-\frac{16790363951822}{389392534508767}a^{6}+\frac{80857456612209}{389392534508767}a^{5}-\frac{193675094535025}{389392534508767}a^{4}+\frac{456195465029423}{389392534508767}a^{3}-\frac{755136736870334}{389392534508767}a^{2}+\frac{530632302910951}{389392534508767}a-\frac{13\!\cdots\!83}{389392534508767}$, $\frac{2879034168043}{389392534508767}a^{9}-\frac{9335514350891}{389392534508767}a^{8}+\frac{131777964165202}{389392534508767}a^{7}-\frac{383411194374604}{389392534508767}a^{6}+\frac{21\!\cdots\!11}{389392534508767}a^{5}-\frac{50\!\cdots\!94}{389392534508767}a^{4}+\frac{12\!\cdots\!07}{389392534508767}a^{3}-\frac{22\!\cdots\!35}{389392534508767}a^{2}+\frac{15\!\cdots\!12}{389392534508767}a-\frac{12\!\cdots\!08}{389392534508767}$ | 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: | \( 152.988253036 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 152.988253036 \cdot 21}{2\cdot\sqrt{3618214860447423}}\cr\approx \mathstrut & 0.261516773741 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{-223}) \), 5.3.18063.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.3.18063.1 |
Degree 6 sibling: | 6.0.898254927.1 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.3.18063.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.10.8.1 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 598 x^{5} + 750 x^{4} + 640 x^{3} + 280 x^{2} + 40 x + 17$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(223\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |