Normalized defining polynomial
\( x^{10} - x^{9} + 3x^{8} - 44x^{7} - 13x^{6} + 71x^{5} + 333x^{4} + 1190x^{3} + 1101x^{2} - 15973x - 13693 \)
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: | \(417483014440192\) \(\medspace = 2^{8}\cdot 277^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(28.98\) | 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: | $2^{4/5}277^{1/2}\approx 28.97769793904893$ | ||
Ramified primes: | \(2\), \(277\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{277}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{81\!\cdots\!85}a^{9}-\frac{25\!\cdots\!59}{16\!\cdots\!77}a^{8}-\frac{66\!\cdots\!57}{81\!\cdots\!85}a^{7}-\frac{31\!\cdots\!21}{81\!\cdots\!85}a^{6}-\frac{98\!\cdots\!49}{81\!\cdots\!85}a^{5}+\frac{26\!\cdots\!77}{81\!\cdots\!85}a^{4}-\frac{76\!\cdots\!40}{16\!\cdots\!77}a^{3}+\frac{24\!\cdots\!85}{16\!\cdots\!77}a^{2}-\frac{33\!\cdots\!74}{81\!\cdots\!85}a+\frac{16\!\cdots\!73}{81\!\cdots\!85}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{59265741677029}{81\!\cdots\!85}a^{9}-\frac{40070029663817}{16\!\cdots\!77}a^{8}+\frac{91693385771437}{81\!\cdots\!85}a^{7}-\frac{26\!\cdots\!64}{81\!\cdots\!85}a^{6}+\frac{39\!\cdots\!64}{81\!\cdots\!85}a^{5}+\frac{81\!\cdots\!93}{81\!\cdots\!85}a^{4}-\frac{143418959613112}{16\!\cdots\!77}a^{3}+\frac{13\!\cdots\!21}{16\!\cdots\!77}a^{2}-\frac{16\!\cdots\!36}{81\!\cdots\!85}a-\frac{71\!\cdots\!68}{81\!\cdots\!85}$, $\frac{34111716853046}{81\!\cdots\!85}a^{9}-\frac{20646456823359}{16\!\cdots\!77}a^{8}+\frac{417444313836153}{81\!\cdots\!85}a^{7}-\frac{26\!\cdots\!21}{81\!\cdots\!85}a^{6}+\frac{27\!\cdots\!56}{81\!\cdots\!85}a^{5}-\frac{13\!\cdots\!93}{81\!\cdots\!85}a^{4}+\frac{22\!\cdots\!27}{16\!\cdots\!77}a^{3}+\frac{25\!\cdots\!41}{16\!\cdots\!77}a^{2}+\frac{57\!\cdots\!76}{81\!\cdots\!85}a+\frac{30\!\cdots\!63}{81\!\cdots\!85}$, $\frac{91514373865214}{81\!\cdots\!85}a^{9}+\frac{31697507355826}{16\!\cdots\!77}a^{8}+\frac{345553631920717}{81\!\cdots\!85}a^{7}-\frac{47\!\cdots\!04}{81\!\cdots\!85}a^{6}-\frac{22\!\cdots\!51}{81\!\cdots\!85}a^{5}-\frac{47\!\cdots\!27}{81\!\cdots\!85}a^{4}-\frac{14\!\cdots\!79}{16\!\cdots\!77}a^{3}+\frac{87\!\cdots\!43}{16\!\cdots\!77}a^{2}+\frac{16\!\cdots\!69}{81\!\cdots\!85}a+\frac{16\!\cdots\!92}{81\!\cdots\!85}$, $\frac{289347171714204}{81\!\cdots\!85}a^{9}-\frac{29294377983142}{16\!\cdots\!77}a^{8}-\frac{17\!\cdots\!03}{81\!\cdots\!85}a^{7}-\frac{11\!\cdots\!59}{81\!\cdots\!85}a^{6}+\frac{17\!\cdots\!09}{81\!\cdots\!85}a^{5}+\frac{23\!\cdots\!03}{81\!\cdots\!85}a^{4}+\frac{37\!\cdots\!00}{16\!\cdots\!77}a^{3}+\frac{90\!\cdots\!25}{16\!\cdots\!77}a^{2}-\frac{11\!\cdots\!91}{81\!\cdots\!85}a-\frac{58\!\cdots\!93}{81\!\cdots\!85}$, $\frac{783111769235826}{81\!\cdots\!85}a^{9}+\frac{414790501316453}{16\!\cdots\!77}a^{8}+\frac{80\!\cdots\!63}{81\!\cdots\!85}a^{7}-\frac{33\!\cdots\!56}{81\!\cdots\!85}a^{6}-\frac{45\!\cdots\!39}{81\!\cdots\!85}a^{5}-\frac{41\!\cdots\!23}{81\!\cdots\!85}a^{4}+\frac{79\!\cdots\!72}{16\!\cdots\!77}a^{3}+\frac{25\!\cdots\!24}{16\!\cdots\!77}a^{2}+\frac{51\!\cdots\!36}{81\!\cdots\!85}a+\frac{35\!\cdots\!43}{81\!\cdots\!85}$ | 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: | \( 678.741594816 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 678.741594816 \cdot 3}{2\cdot\sqrt{417483014440192}}\cr\approx \mathstrut & 0.310638859506 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{277}) \), 5.1.4432.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.4432.1 |
Degree 6 sibling: | 6.2.340062928.2 |
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: | 5.1.4432.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(277\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |