Normalized defining polynomial
\( x^{10} - x^{9} - 2x^{8} + 4x^{7} + 4x^{6} - 7x^{5} - 25x^{4} + 34x^{3} - 4x^{2} - 24x + 16 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-14347135602899\) \(\medspace = -\,11^{3}\cdot 47^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(20.69\) | 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))
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Galois root discriminant: | $11^{1/2}47^{2/3}\approx 43.19447977961657$ | ||
Ramified primes: | \(11\), \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{8}-\frac{1}{2}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{3104}a^{9}+\frac{89}{1552}a^{8}+\frac{11}{776}a^{7}+\frac{15}{388}a^{6}-\frac{61}{776}a^{5}-\frac{227}{3104}a^{4}-\frac{153}{1552}a^{3}+\frac{89}{776}a^{2}+\frac{11}{388}a+\frac{13}{194}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{185}{3104}a^{9}-\frac{25}{1552}a^{8}-\frac{99}{776}a^{7}+\frac{59}{388}a^{6}+\frac{355}{776}a^{5}-\frac{91}{3104}a^{4}-\frac{2115}{1552}a^{3}+\frac{751}{776}a^{2}+\frac{95}{388}a-\frac{117}{194}$, $\frac{3}{3104}a^{9}-\frac{121}{1552}a^{8}+\frac{33}{776}a^{7}+\frac{45}{388}a^{6}-\frac{183}{776}a^{5}-\frac{681}{3104}a^{4}+\frac{705}{1552}a^{3}+\frac{1043}{776}a^{2}-\frac{743}{388}a+\frac{233}{194}$, $\frac{49}{3104}a^{9}+\frac{93}{1552}a^{8}-\frac{43}{776}a^{7}-\frac{41}{388}a^{6}+\frac{115}{776}a^{5}+\frac{1293}{3104}a^{4}-\frac{901}{1552}a^{3}-\frac{877}{776}a^{2}+\frac{151}{388}a+\frac{55}{194}$, $\frac{437}{3104}a^{9}-\frac{101}{1552}a^{8}-\frac{237}{776}a^{7}+\frac{153}{388}a^{6}+\frac{503}{776}a^{5}-\frac{1423}{3104}a^{4}-\frac{5751}{1552}a^{3}+\frac{1645}{776}a^{2}+\frac{345}{388}a-\frac{527}{194}$, $\frac{619}{3104}a^{9}-\frac{5}{1552}a^{8}-\frac{369}{776}a^{7}+\frac{167}{388}a^{6}+\frac{1041}{776}a^{5}-\frac{833}{3104}a^{4}-\frac{8571}{1552}a^{3}+\frac{1353}{776}a^{2}+\frac{795}{388}a-\frac{877}{194}$, $\frac{277}{3104}a^{9}-\frac{179}{1552}a^{8}-\frac{57}{776}a^{7}+\frac{81}{388}a^{6}+\frac{175}{776}a^{5}-\frac{799}{3104}a^{4}-\frac{3581}{1552}a^{3}+\frac{2149}{776}a^{2}-\frac{639}{388}a+\frac{109}{194}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 648.968901126 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 648.968901126 \cdot 1}{2\cdot\sqrt{14347135602899}}\cr\approx \mathstrut & 0.339993705083 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.3.24299.1 |
Degree 6 sibling: | 6.0.6494855411.1 |
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: | 5.3.24299.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | R | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.3.2.1 | $x^{3} + 47$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
47.6.4.1 | $x^{6} + 135 x^{5} + 6090 x^{4} + 92569 x^{3} + 36795 x^{2} + 287490 x + 4253484$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |