Normalized defining polynomial
\( x^{10} - 2x^{9} - 19x^{8} + 12x^{7} + 101x^{6} - 10x^{5} - 175x^{4} - 16x^{3} + 74x^{2} + 24x + 2 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(3688067268608000\) \(\medspace = 2^{18}\cdot 5^{3}\cdot 103^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(36.03\) | 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: | $2^{13/6}5^{1/2}103^{1/2}\approx 101.89087030315206$ | ||
Ramified primes: | \(2\), \(5\), \(103\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{6}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{5}{16}a^{5}-\frac{7}{16}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a+\frac{1}{8}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $a^{9}-\frac{17}{8}a^{8}-\frac{75}{4}a^{7}+\frac{29}{2}a^{6}+\frac{397}{4}a^{5}-\frac{201}{8}a^{4}-171a^{3}+16a^{2}+68a+\frac{43}{4}$, $\frac{33}{8}a^{9}-9a^{8}-77a^{7}+\frac{257}{4}a^{6}+\frac{3269}{8}a^{5}-\frac{481}{4}a^{4}-713a^{3}+69a^{2}+\frac{1201}{4}a+\frac{89}{2}$, $\frac{55}{16}a^{9}-\frac{117}{16}a^{8}-\frac{515}{8}a^{7}+\frac{397}{8}a^{6}+\frac{5445}{16}a^{5}-\frac{1291}{16}a^{4}-589a^{3}+\frac{63}{2}a^{2}+\frac{2011}{8}a+\frac{341}{8}$, $\frac{43}{16}a^{9}-\frac{89}{16}a^{8}-\frac{409}{8}a^{7}+\frac{303}{8}a^{6}+\frac{4365}{16}a^{5}-\frac{1019}{16}a^{4}-\frac{951}{2}a^{3}+30a^{2}+\frac{1603}{8}a+\frac{269}{8}$, $\frac{25}{16}a^{9}-\frac{55}{16}a^{8}-\frac{231}{8}a^{7}+\frac{193}{8}a^{6}+\frac{2423}{16}a^{5}-\frac{693}{16}a^{4}-\frac{521}{2}a^{3}+23a^{2}+\frac{897}{8}a+\frac{147}{8}$, $\frac{5}{8}a^{9}-\frac{11}{8}a^{8}-\frac{23}{2}a^{7}+\frac{19}{2}a^{6}+\frac{477}{8}a^{5}-\frac{119}{8}a^{4}-\frac{203}{2}a^{3}+\frac{1}{2}a^{2}+\frac{187}{4}a+\frac{33}{4}$, $\frac{1}{16}a^{9}-\frac{3}{16}a^{8}-\frac{7}{8}a^{7}+\frac{9}{8}a^{6}+\frac{63}{16}a^{5}-\frac{17}{16}a^{4}-\frac{11}{2}a^{3}-2a^{2}+\frac{17}{8}a+\frac{7}{8}$, $\frac{15}{16}a^{9}-\frac{37}{16}a^{8}-\frac{131}{8}a^{7}+\frac{141}{8}a^{6}+\frac{1309}{16}a^{5}-\frac{587}{16}a^{4}-131a^{3}+\frac{43}{2}a^{2}+\frac{395}{8}a+\frac{53}{8}$, $\frac{5}{4}a^{9}-\frac{15}{8}a^{8}-24a^{7}+\frac{5}{4}a^{6}+\frac{465}{4}a^{5}+\frac{439}{8}a^{4}-\frac{293}{2}a^{3}-\frac{183}{2}a^{2}-a+\frac{7}{4}$ | 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: | \( 159816.79149 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 159816.79149 \cdot 1}{2\cdot\sqrt{3688067268608000}}\cr\approx \mathstrut & 1.3473888575 \end{aligned}\]
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: | 5.5.13579520.1 |
Degree 6 sibling: | 6.6.1357952000.1 |
Degree 10 sibling: | deg 10 |
Degree 12 sibling: | 12.12.1844033634304000000.1 |
Degree 15 sibling: | deg 15 |
Degree 20 siblings: | 20.20.8501150111111046013911040000000000.1, deg 20, deg 20 |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.5.13579520.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.8 | $x^{4} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ |
2.6.10.2 | $x^{6} + 2 x^{5} + 4 x^{4} + 4 x + 2$ | $6$ | $1$ | $10$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(103\) | $\Q_{103}$ | $x + 98$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{103}$ | $x + 98$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
103.2.1.1 | $x^{2} + 309$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |