Normalized defining polynomial
\( x^{10} - 3x^{9} + 2x^{8} - 5x^{7} + 23x^{6} - 19x^{5} - 45x^{4} + 109x^{3} - 104x^{2} + 67x - 25 \)
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: | \(8123212515625\) \(\medspace = 5^{6}\cdot 151^{4}\) | 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: | \(19.54\) | 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: | $5^{2/3}151^{1/2}\approx 35.93093151785668$ | ||
Ramified primes: | \(5\), \(151\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{165}a^{8}+\frac{23}{165}a^{7}+\frac{7}{55}a^{6}-\frac{16}{165}a^{5}-\frac{67}{165}a^{4}+\frac{26}{55}a^{3}-\frac{34}{165}a^{2}+\frac{43}{165}a+\frac{5}{11}$, $\frac{1}{825}a^{9}-\frac{41}{275}a^{7}-\frac{224}{825}a^{6}+\frac{301}{825}a^{5}+\frac{409}{825}a^{4}-\frac{68}{825}a^{3}+\frac{2}{5}a^{2}+\frac{62}{275}a-\frac{14}{33}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{163}{825}a^{9}-\frac{76}{165}a^{8}+\frac{86}{825}a^{7}-\frac{767}{825}a^{6}+\frac{96}{25}a^{5}-\frac{366}{275}a^{4}-\frac{7724}{825}a^{3}+\frac{2617}{165}a^{2}-\frac{9122}{825}a+\frac{175}{33}$, $\frac{1}{11}a^{9}-\frac{37}{165}a^{8}-\frac{1}{165}a^{7}-\frac{67}{165}a^{6}+\frac{322}{165}a^{5}-\frac{7}{55}a^{4}-\frac{826}{165}a^{3}+\frac{1093}{165}a^{2}-\frac{76}{15}a+\frac{89}{33}$, $\frac{116}{825}a^{9}-\frac{41}{165}a^{8}-\frac{8}{825}a^{7}-\frac{589}{825}a^{6}+\frac{632}{275}a^{5}+\frac{43}{275}a^{4}-\frac{4903}{825}a^{3}+\frac{259}{33}a^{2}-\frac{4564}{825}a+\frac{104}{33}$, $\frac{14}{825}a^{9}-\frac{1}{11}a^{8}+\frac{128}{825}a^{7}-\frac{12}{275}a^{6}+\frac{464}{825}a^{5}-\frac{358}{275}a^{4}-\frac{159}{275}a^{3}+\frac{258}{55}a^{2}-\frac{4196}{825}a+\frac{52}{33}$, $\frac{34}{825}a^{9}-\frac{1}{15}a^{8}+\frac{53}{825}a^{7}-\frac{82}{275}a^{6}+\frac{389}{825}a^{5}-\frac{284}{825}a^{4}-\frac{184}{275}a^{3}+\frac{58}{15}a^{2}-\frac{3191}{825}a+\frac{41}{33}$ | 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: | \( 269.012095622 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 269.012095622 \cdot 2}{2\cdot\sqrt{8123212515625}}\cr\approx \mathstrut & 0.588419917564 \end{aligned}\]
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: | 5.1.570025.1 |
Degree 6 sibling: | 6.2.14250625.2 |
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: | 5.1.570025.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(151\) | $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |