Normalized defining polynomial
\( x^{10} - 15x^{8} - 25x^{6} - 196x^{5} + 185x^{4} + 80x^{3} - 180x^{2} - 140x - 22 \)
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: | \(10000000000000000\) \(\medspace = 2^{16}\cdot 5^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.81\) | 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: | $2^{2}5^{8/5}\approx 52.53055608807534$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{80}a^{8}-\frac{1}{40}a^{7}-\frac{1}{40}a^{6}-\frac{7}{40}a^{5}-\frac{3}{16}a^{4}+\frac{7}{20}a^{3}-\frac{3}{40}a^{2}+\frac{3}{10}a+\frac{19}{40}$, $\frac{1}{7884080}a^{9}-\frac{36703}{7884080}a^{8}-\frac{23809}{394204}a^{7}+\frac{61788}{492755}a^{6}-\frac{3525901}{7884080}a^{5}-\frac{2481157}{7884080}a^{4}+\frac{1898023}{3942040}a^{3}+\frac{197391}{788408}a^{2}+\frac{1364227}{3942040}a-\frac{219979}{3942040}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{10443119}{7884080}a^{9}-\frac{2545989}{7884080}a^{8}-\frac{9817101}{492755}a^{7}+\frac{9822369}{1971020}a^{6}-\frac{255996511}{7884080}a^{5}-\frac{1999329083}{7884080}a^{4}+\frac{1230378689}{3942040}a^{3}+\frac{207726241}{3942040}a^{2}-\frac{1167184591}{3942040}a-\frac{292201429}{3942040}$, $\frac{1426983}{7884080}a^{9}-\frac{416507}{7884080}a^{8}-\frac{5394861}{1971020}a^{7}+\frac{78748}{98551}a^{6}-\frac{34009651}{7884080}a^{5}-\frac{267083281}{7884080}a^{4}+\frac{177167117}{3942040}a^{3}+\frac{30263019}{3942040}a^{2}-\frac{27597915}{788408}a-\frac{74772999}{3942040}$, $\frac{31292993}{7884080}a^{9}-\frac{16023103}{7884080}a^{8}-\frac{57916519}{985510}a^{7}+\frac{59842483}{1971020}a^{6}-\frac{874200057}{7884080}a^{5}-\frac{5716749761}{7884080}a^{4}+\frac{4395077583}{3942040}a^{3}-\frac{820513173}{3942040}a^{2}-\frac{2783856537}{3942040}a-\frac{526028143}{3942040}$, $\frac{743513}{1576816}a^{9}+\frac{436529}{7884080}a^{8}-\frac{7474891}{985510}a^{7}-\frac{3806137}{1971020}a^{6}-\frac{45190421}{7884080}a^{5}-\frac{124826045}{1576816}a^{4}+\frac{443997611}{3942040}a^{3}+\frac{814686083}{3942040}a^{2}+\frac{403427043}{3942040}a+\frac{54616321}{3942040}$, $\frac{390411}{7884080}a^{9}+\frac{651773}{7884080}a^{8}-\frac{1599813}{1971020}a^{7}-\frac{1099273}{985510}a^{6}-\frac{83323}{1576816}a^{5}-\frac{110772017}{7884080}a^{4}-\frac{34937743}{3942040}a^{3}+\frac{143528227}{3942040}a^{2}-\frac{65629171}{3942040}a-\frac{21987067}{788408}$ (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: | \( 262016.362387 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 262016.362387 \cdot 1}{2\cdot\sqrt{10000000000000000}}\cr\approx \mathstrut & 8.16728822277 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 60 |
The 5 conjugacy class representatives for $A_{5}$ |
Character table for $A_{5}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.1.100000000.1 |
Degree 6 sibling: | 6.2.100000000.1 |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Minimal sibling: | 5.1.100000000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.4.8.1 | $x^{4} + 2 x^{2} + 4 x + 10$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
\(5\) | 5.5.8.6 | $x^{5} + 5 x^{4} + 5$ | $5$ | $1$ | $8$ | $D_{5}$ | $[2]^{2}$ |
5.5.8.6 | $x^{5} + 5 x^{4} + 5$ | $5$ | $1$ | $8$ | $D_{5}$ | $[2]^{2}$ |