Normalized defining polynomial
\( x^{10} - 2x^{9} + x^{8} + 4x^{7} - 6x^{6} + 4x^{5} + 2x^{4} - 4x^{3} + 6x^{2} + 2 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-24067178496\) \(\medspace = -\,2^{18}\cdot 3^{2}\cdot 101^{2}\) | 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: | \(10.92\) | 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: | $2^{103/48}3^{1/2}101^{2/3}\approx 166.2398259737741$ | ||
Ramified primes: | \(2\), \(3\), \(101\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{1}{2} a^{9} + \frac{3}{2} a^{8} - 2 a^{7} + 3 a^{5} - 4 a^{4} + 2 a^{3} - a \) (order $4$) | 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: | $\frac{2}{3}a^{9}-\frac{5}{3}a^{8}+\frac{5}{3}a^{7}+\frac{2}{3}a^{6}-\frac{8}{3}a^{5}+\frac{8}{3}a^{4}-\frac{5}{3}a^{3}+\frac{1}{3}a^{2}+\frac{2}{3}a+1$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{2}{3}a^{6}+\frac{7}{3}a^{5}-\frac{4}{3}a^{4}-\frac{5}{3}a^{3}+\frac{8}{3}a^{2}-\frac{4}{3}a+\frac{1}{3}$, $\frac{2}{3}a^{9}-\frac{5}{3}a^{8}+\frac{5}{3}a^{7}+\frac{2}{3}a^{6}-\frac{5}{3}a^{5}+\frac{5}{3}a^{4}-\frac{5}{3}a^{3}+\frac{10}{3}a^{2}-\frac{1}{3}a+1$, $\frac{2}{3}a^{8}-\frac{5}{3}a^{7}+\frac{5}{3}a^{6}+\frac{2}{3}a^{5}-\frac{8}{3}a^{4}+\frac{8}{3}a^{3}-\frac{5}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$ | 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: | \( 59.5403337985 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 59.5403337985 \cdot 1}{4\cdot\sqrt{24067178496}}\cr\approx \mathstrut & 0.939589221381 \end{aligned}\]
Galois group
$\PGOPlus(4,5)$ (as 10T41):
A non-solvable group of order 14400 |
The 25 conjugacy class representatives for $(A_5^2 : C_2):C_2$ |
Character table for $(A_5^2 : C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 25 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.8.16.64 | $x^{8} + 2 x^{4} + 4 x + 2$ | $8$ | $1$ | $16$ | $V_4^2:(S_3\times C_2)$ | $[4/3, 4/3, 2, 7/3, 7/3]_{3}^{2}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(101\) | $\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
101.3.2.1 | $x^{3} + 101$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |