Normalized defining polynomial
\( x^{10} - 2x^{9} + 3x^{8} - 2x^{7} - 23x^{6} - 61x^{4} + 236x^{3} + 186x^{2} - 568x - 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9227446944279201\) \(\medspace = 3^{16}\cdot 11^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.49\) | 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: | $3^{11/6}11^{4/5}\approx 51.031278439357436$ | ||
Ramified primes: | \(3\), \(11\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{14}a^{8}-\frac{1}{14}a^{7}-\frac{3}{14}a^{6}-\frac{1}{14}a^{4}+\frac{3}{7}a^{3}-\frac{1}{14}a^{2}+\frac{1}{7}a-\frac{3}{14}$, $\frac{1}{13183842}a^{9}+\frac{33589}{2197307}a^{8}+\frac{274590}{2197307}a^{7}-\frac{1147847}{13183842}a^{6}-\frac{1050355}{4394614}a^{5}+\frac{2013793}{4394614}a^{4}+\frac{3065105}{13183842}a^{3}-\frac{1541303}{4394614}a^{2}-\frac{102125}{627802}a+\frac{2756669}{13183842}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{10}$, which has order $10$
Unit group
Rank: | $5$ | 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)
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Fundamental units: | $\frac{5897}{1883406}a^{9}+\frac{2802}{313901}a^{8}-\frac{2157}{627802}a^{7}+\frac{107405}{1883406}a^{6}-\frac{48903}{627802}a^{5}-\frac{165311}{627802}a^{4}-\frac{532450}{941703}a^{3}-\frac{344069}{313901}a^{2}+\frac{27604}{44843}a+\frac{399907}{1883406}$, $\frac{25971}{2197307}a^{9}-\frac{205421}{4394614}a^{8}+\frac{518159}{4394614}a^{7}-\frac{498170}{2197307}a^{6}+\frac{192793}{2197307}a^{5}-\frac{185365}{4394614}a^{4}-\frac{1137744}{2197307}a^{3}+\frac{7529050}{2197307}a^{2}-\frac{356021}{89686}a+\frac{166123}{2197307}$, $\frac{369583}{13183842}a^{9}-\frac{233361}{2197307}a^{8}+\frac{544373}{2197307}a^{7}-\frac{6370361}{13183842}a^{6}+\frac{481111}{4394614}a^{5}-\frac{635097}{4394614}a^{4}-\frac{14689447}{13183842}a^{3}+\frac{41959795}{4394614}a^{2}-\frac{785587}{89686}a-\frac{14243485}{13183842}$, $\frac{45197}{4394614}a^{9}-\frac{23510}{2197307}a^{8}+\frac{135080}{2197307}a^{7}-\frac{194787}{4394614}a^{6}-\frac{115800}{2197307}a^{5}-\frac{898955}{2197307}a^{4}-\frac{2856981}{2197307}a^{3}-\frac{2300}{2197307}a^{2}-\frac{699021}{627802}a-\frac{51113}{2197307}$, $\frac{27877}{4394614}a^{9}-\frac{18390}{2197307}a^{8}+\frac{47765}{4394614}a^{7}-\frac{90681}{4394614}a^{6}-\frac{248533}{2197307}a^{5}+\frac{208080}{2197307}a^{4}-\frac{1059125}{4394614}a^{3}+\frac{3990061}{4394614}a^{2}+\frac{103060}{313901}a-\frac{7539425}{2197307}$ | 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: | \( 4040.85411722 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 4040.85411722 \cdot 10}{2\cdot\sqrt{9227446944279201}}\cr\approx \mathstrut & 1.31123901398 \end{aligned}\]
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: | 5.1.10673289.2 |
Degree 6 sibling: | 6.2.864536409.1 |
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: | 5.1.10673289.2 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.5.1 | $x^{3} + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
3.6.11.9 | $x^{6} + 3$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |