Normalized defining polynomial
\( x^{10} - 5x^{9} + 11x^{8} - 14x^{7} + 33x^{6} - 71x^{5} + 101x^{4} - 90x^{3} + 107x^{2} - 73x - 11 \)
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: | \(10628820000000\) \(\medspace = 2^{8}\cdot 3^{12}\cdot 5^{7}\) | 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: | \(20.07\) | 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: | $2^{4/5}3^{11/6}5^{3/4}\approx 43.628853456312655$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{204}a^{8}-\frac{1}{51}a^{7}+\frac{7}{102}a^{5}+\frac{47}{204}a^{4}+\frac{41}{102}a^{3}+\frac{29}{102}a^{2}+\frac{1}{34}a-\frac{89}{204}$, $\frac{1}{5100}a^{9}+\frac{2}{1275}a^{8}+\frac{19}{510}a^{7}+\frac{38}{425}a^{6}-\frac{839}{5100}a^{5}-\frac{17}{150}a^{4}+\frac{101}{425}a^{3}+\frac{283}{2550}a^{2}-\frac{7}{20}a+\frac{911}{2550}$
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)
| |
Fundamental units: | $\frac{22}{1275}a^{9}-\frac{33}{425}a^{8}+\frac{12}{85}a^{7}-\frac{56}{425}a^{6}+\frac{214}{425}a^{5}-\frac{472}{425}a^{4}+\frac{92}{75}a^{3}-\frac{316}{425}a^{2}+\frac{162}{85}a-\frac{1741}{1275}$, $\frac{7}{425}a^{9}-\frac{203}{5100}a^{8}-\frac{3}{170}a^{7}+\frac{226}{1275}a^{6}+\frac{287}{2550}a^{5}-\frac{427}{5100}a^{4}-\frac{657}{850}a^{3}+\frac{937}{425}a^{2}-\frac{152}{255}a+\frac{2273}{5100}$, $\frac{11}{1275}a^{9}-\frac{33}{850}a^{8}+\frac{6}{85}a^{7}-\frac{28}{425}a^{6}+\frac{107}{425}a^{5}-\frac{236}{425}a^{4}+\frac{46}{75}a^{3}+\frac{109}{850}a^{2}-\frac{4}{85}a-\frac{233}{1275}$, $\frac{227}{1700}a^{9}-\frac{3427}{5100}a^{8}+\frac{382}{255}a^{7}-\frac{4957}{2550}a^{6}+\frac{7547}{1700}a^{5}-\frac{48293}{5100}a^{4}+\frac{17818}{1275}a^{3}-\frac{34477}{2550}a^{2}+\frac{15853}{1020}a-\frac{18981}{1700}$, $\frac{127}{340}a^{9}-\frac{619}{340}a^{8}+\frac{395}{102}a^{7}-\frac{2297}{510}a^{6}+\frac{10831}{1020}a^{5}-\frac{7791}{340}a^{4}+\frac{17201}{510}a^{3}-\frac{7691}{255}a^{2}+\frac{7901}{204}a-\frac{25783}{1020}$ | 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: | \( 755.722725676 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 755.722725676 \cdot 1}{2\cdot\sqrt{10628820000000}}\cr\approx \mathstrut & 0.722552528636 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.1458000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.1458000.1 |
Degree 6 sibling: | 6.2.118098000.6 |
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: | 5.1.1458000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.10.1 | $x^{6} + 18 x^{4} + 6 x^{3} + 162 x^{2} + 216 x + 90$ | $3$ | $2$ | $10$ | $D_{6}$ | $[5/2]_{2}^{2}$ | |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |