Normalized defining polynomial
\( x^{10} - 2x^{9} + 3x^{8} + 5x^{7} + 5x^{6} + 20x^{5} + 29x^{4} + 43x^{3} + 27x^{2} + 27x + 11 \)
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: | \(25347992633041\) \(\medspace = 13^{4}\cdot 31^{6}\) | 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: | \(21.90\) | 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: | $13^{1/2}31^{2/3}\approx 35.58056215005226$ | ||
Ramified primes: | \(13\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13}a^{8}+\frac{2}{13}a^{6}-\frac{4}{13}a^{5}-\frac{5}{13}a^{4}+\frac{1}{13}a^{3}-\frac{3}{13}a^{2}-\frac{3}{13}a-\frac{2}{13}$, $\frac{1}{25883}a^{9}+\frac{201}{25883}a^{8}+\frac{4968}{25883}a^{7}-\frac{928}{25883}a^{6}-\frac{1225}{25883}a^{5}-\frac{161}{2353}a^{4}+\frac{887}{25883}a^{3}-\frac{11032}{25883}a^{2}-\frac{9549}{25883}a+\frac{617}{2353}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{193}{25883}a^{9}-\frac{79}{1991}a^{8}+\frac{1153}{25883}a^{7}+\frac{86}{25883}a^{6}+\frac{504}{25883}a^{5}-\frac{1208}{2353}a^{4}+\frac{1956}{25883}a^{3}-\frac{16725}{25883}a^{2}-\frac{15219}{25883}a-\frac{57}{181}$, $\frac{1084}{25883}a^{9}-\frac{1126}{25883}a^{8}+\frac{1648}{25883}a^{7}+\frac{5476}{25883}a^{6}+\frac{14034}{25883}a^{5}+\frac{2675}{2353}a^{4}+\frac{43657}{25883}a^{3}+\frac{60979}{25883}a^{2}+\frac{37922}{25883}a+\frac{2748}{2353}$, $\frac{329}{2353}a^{9}-\frac{841}{2353}a^{8}+\frac{1490}{2353}a^{7}+\frac{759}{2353}a^{6}+\frac{1329}{2353}a^{5}+\frac{6315}{2353}a^{4}+\frac{6024}{2353}a^{3}+\frac{11468}{2353}a^{2}+\frac{2892}{2353}a+\frac{6804}{2353}$, $\frac{193}{25883}a^{9}-\frac{79}{1991}a^{8}+\frac{1153}{25883}a^{7}+\frac{86}{25883}a^{6}+\frac{504}{25883}a^{5}-\frac{1208}{2353}a^{4}+\frac{1956}{25883}a^{3}-\frac{16725}{25883}a^{2}-\frac{15219}{25883}a-\frac{238}{181}$, $\frac{252}{25883}a^{9}+\frac{8841}{25883}a^{8}-\frac{16331}{25883}a^{7}+\frac{19001}{25883}a^{6}+\frac{65608}{25883}a^{5}+\frac{332}{181}a^{4}+\frac{155830}{25883}a^{3}+\frac{270148}{25883}a^{2}+\frac{25636}{1991}a+\frac{10141}{2353}$ | 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: | \( 472.17814678 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 472.17814678 \cdot 1}{2\cdot\sqrt{25347992633041}}\cr\approx \mathstrut & 0.29233685220 \end{aligned}\]
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: | 5.1.162409.1 |
Degree 6 sibling: | 6.2.156075049.3 |
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: | 5.1.162409.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |