Normalized defining polynomial
\( x^{10} - 2 x^{9} + 29 x^{8} - 15 x^{7} + 205 x^{6} - 1592 x^{5} - 1518 x^{4} - 14731 x^{3} + \cdots - 4099849 \)
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: | \(10784008474947049\) \(\medspace = 1609^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.11\) | 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: | $1609^{1/2}\approx 40.11234224026316$ | ||
Ramified primes: | \(1609\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1609}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{8}a^{8}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{18\!\cdots\!84}a^{9}-\frac{10\!\cdots\!81}{18\!\cdots\!84}a^{8}-\frac{20\!\cdots\!63}{42\!\cdots\!36}a^{7}+\frac{50\!\cdots\!53}{18\!\cdots\!84}a^{6}-\frac{13\!\cdots\!91}{94\!\cdots\!92}a^{5}-\frac{24\!\cdots\!35}{94\!\cdots\!92}a^{4}-\frac{16\!\cdots\!01}{47\!\cdots\!96}a^{3}-\frac{37\!\cdots\!07}{18\!\cdots\!84}a^{2}+\frac{48\!\cdots\!85}{18\!\cdots\!84}a+\frac{73\!\cdots\!85}{18\!\cdots\!84}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (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: | $\frac{15\!\cdots\!95}{18\!\cdots\!84}a^{9}+\frac{18\!\cdots\!57}{18\!\cdots\!84}a^{8}+\frac{25\!\cdots\!97}{21\!\cdots\!18}a^{7}+\frac{31\!\cdots\!55}{18\!\cdots\!84}a^{6}-\frac{43\!\cdots\!65}{94\!\cdots\!92}a^{5}-\frac{22\!\cdots\!79}{94\!\cdots\!92}a^{4}-\frac{18\!\cdots\!23}{47\!\cdots\!96}a^{3}+\frac{66\!\cdots\!31}{18\!\cdots\!84}a^{2}-\frac{35\!\cdots\!25}{18\!\cdots\!84}a+\frac{52\!\cdots\!75}{18\!\cdots\!84}$, $\frac{59\!\cdots\!85}{18\!\cdots\!84}a^{9}+\frac{29\!\cdots\!09}{18\!\cdots\!84}a^{8}+\frac{24\!\cdots\!98}{10\!\cdots\!59}a^{7}-\frac{13\!\cdots\!79}{18\!\cdots\!84}a^{6}+\frac{53\!\cdots\!62}{11\!\cdots\!49}a^{5}-\frac{27\!\cdots\!76}{11\!\cdots\!49}a^{4}-\frac{33\!\cdots\!03}{94\!\cdots\!92}a^{3}-\frac{22\!\cdots\!97}{18\!\cdots\!84}a^{2}+\frac{94\!\cdots\!01}{18\!\cdots\!84}a+\frac{69\!\cdots\!71}{18\!\cdots\!84}$, $\frac{17\!\cdots\!75}{18\!\cdots\!84}a^{9}+\frac{17\!\cdots\!95}{18\!\cdots\!84}a^{8}+\frac{56\!\cdots\!04}{10\!\cdots\!59}a^{7}+\frac{50\!\cdots\!87}{18\!\cdots\!84}a^{6}+\frac{34\!\cdots\!29}{47\!\cdots\!96}a^{5}-\frac{11\!\cdots\!51}{47\!\cdots\!96}a^{4}-\frac{18\!\cdots\!55}{94\!\cdots\!92}a^{3}-\frac{17\!\cdots\!31}{18\!\cdots\!84}a^{2}-\frac{65\!\cdots\!13}{18\!\cdots\!84}a-\frac{19\!\cdots\!43}{18\!\cdots\!84}$, $\frac{13\!\cdots\!89}{23\!\cdots\!98}a^{9}+\frac{28\!\cdots\!35}{47\!\cdots\!96}a^{8}+\frac{42\!\cdots\!05}{42\!\cdots\!36}a^{7}+\frac{22\!\cdots\!33}{47\!\cdots\!96}a^{6}+\frac{23\!\cdots\!45}{11\!\cdots\!49}a^{5}-\frac{71\!\cdots\!51}{47\!\cdots\!96}a^{4}+\frac{12\!\cdots\!61}{11\!\cdots\!49}a^{3}+\frac{79\!\cdots\!65}{47\!\cdots\!96}a^{2}-\frac{34\!\cdots\!07}{47\!\cdots\!96}a-\frac{12\!\cdots\!30}{11\!\cdots\!49}$, $\frac{50\!\cdots\!23}{47\!\cdots\!96}a^{9}+\frac{79\!\cdots\!03}{47\!\cdots\!96}a^{8}+\frac{23\!\cdots\!37}{21\!\cdots\!18}a^{7}+\frac{28\!\cdots\!75}{47\!\cdots\!96}a^{6}+\frac{16\!\cdots\!85}{23\!\cdots\!98}a^{5}-\frac{49\!\cdots\!81}{23\!\cdots\!98}a^{4}-\frac{23\!\cdots\!57}{23\!\cdots\!98}a^{3}-\frac{22\!\cdots\!11}{47\!\cdots\!96}a^{2}-\frac{10\!\cdots\!63}{47\!\cdots\!96}a-\frac{31\!\cdots\!87}{47\!\cdots\!96}$ (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: | \( 3525.875783 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 3525.875783 \cdot 3}{2\cdot\sqrt{10784008474947049}}\cr\approx \mathstrut & 0.3175028453 \end{aligned}\] (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
\(\Q(\sqrt{1609}) \), 5.1.1609.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.1609.1 |
Degree 6 sibling: | 6.2.4165509529.1 |
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: | 5.1.1609.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(1609\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |