Normalized defining polynomial
\( x^{10} + 14x^{8} - 8x^{7} + 39x^{6} - 132x^{5} - 858x^{4} + 312x^{3} - 3892x^{2} + 2390x - 3581 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(205050250625625\) \(\medspace = 3^{5}\cdot 5^{4}\cdot 67^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(26.99\) | 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: | $3^{1/2}5^{2/3}67^{1/2}\approx 41.45510615605851$ | ||
Ramified primes: | \(3\), \(5\), \(67\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{201}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{910749017532642}a^{9}+\frac{64823539536569}{455374508766321}a^{8}+\frac{1943150729993}{303583005844214}a^{7}+\frac{75615899988494}{455374508766321}a^{6}-\frac{45820892942867}{910749017532642}a^{5}+\frac{370826877648547}{910749017532642}a^{4}-\frac{448448819775947}{910749017532642}a^{3}-\frac{126966567544411}{455374508766321}a^{2}+\frac{33324857694058}{151791502922107}a+\frac{79756494477089}{910749017532642}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $5$ | 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: | $\frac{47234421983}{910749017532642}a^{9}+\frac{148719239371}{455374508766321}a^{8}+\frac{216061248749}{151791502922107}a^{7}+\frac{1496365609483}{455374508766321}a^{6}+\frac{2995679253781}{455374508766321}a^{5}-\frac{12011631202321}{910749017532642}a^{4}-\frac{86685741065071}{910749017532642}a^{3}-\frac{422938321331293}{910749017532642}a^{2}-\frac{80028441543235}{151791502922107}a-\frac{506121988622651}{910749017532642}$, $\frac{89815740889}{151791502922107}a^{9}+\frac{248197546377}{303583005844214}a^{8}+\frac{2471828112719}{303583005844214}a^{7}+\frac{1669962833355}{303583005844214}a^{6}+\frac{3948762144221}{303583005844214}a^{5}-\frac{7459318168197}{151791502922107}a^{4}-\frac{189279959401633}{303583005844214}a^{3}-\frac{146190698688829}{303583005844214}a^{2}-\frac{257372948062312}{151791502922107}a-\frac{115592608391756}{151791502922107}$, $\frac{1367927441815}{455374508766321}a^{9}-\frac{295511885687}{455374508766321}a^{8}+\frac{4370211948516}{151791502922107}a^{7}-\frac{21637770595322}{455374508766321}a^{6}+\frac{41448809764079}{910749017532642}a^{5}-\frac{93859565658731}{455374508766321}a^{4}-\frac{18\!\cdots\!41}{910749017532642}a^{3}+\frac{681986783210356}{455374508766321}a^{2}-\frac{976767011343628}{151791502922107}a+\frac{56\!\cdots\!19}{910749017532642}$, $\frac{77429794190}{151791502922107}a^{9}+\frac{1328735648891}{303583005844214}a^{8}+\frac{6487537404053}{303583005844214}a^{7}+\frac{14438802634180}{151791502922107}a^{6}+\frac{83794199079411}{303583005844214}a^{5}+\frac{231940959956423}{303583005844214}a^{4}+\frac{400094114200579}{303583005844214}a^{3}+\frac{389982018929377}{303583005844214}a^{2}+\frac{562831143304117}{303583005844214}a-\frac{246967156984235}{151791502922107}$, $\frac{18477010374979}{910749017532642}a^{9}-\frac{35503217265175}{455374508766321}a^{8}+\frac{40612387955292}{151791502922107}a^{7}-\frac{146306152427869}{455374508766321}a^{6}-\frac{10\!\cdots\!35}{910749017532642}a^{5}+\frac{30\!\cdots\!91}{910749017532642}a^{4}-\frac{44\!\cdots\!63}{455374508766321}a^{3}+\frac{13\!\cdots\!77}{910749017532642}a^{2}-\frac{19\!\cdots\!68}{151791502922107}a+\frac{32\!\cdots\!36}{455374508766321}$ (assuming GRH) | 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: | \( 2821.27174753 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 2821.27174753 \cdot 1}{2\cdot\sqrt{205050250625625}}\cr\approx \mathstrut & 0.614135548455 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{201}) \), 5.1.5025.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.5025.1 |
Degree 6 sibling: | 6.2.5075375625.4 |
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: | 5.1.5025.1 |
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 | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(67\) | 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |