Normalized defining polynomial
\( x^{12} + 4x^{10} + 10x^{8} - 10x^{6} + 17x^{4} + 10x^{2} + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(4452139149819904\) \(\medspace = 2^{18}\cdot 19^{8}\) | 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: | \(20.14\) | 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: | $2^{3/2}19^{4/5}\approx 29.822762797001236$ | ||
Ramified primes: | \(2\), \(19\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{10}a^{8}+\frac{1}{10}a^{6}-\frac{1}{2}a^{4}-\frac{1}{5}a^{2}+\frac{3}{10}$, $\frac{1}{20}a^{9}+\frac{1}{20}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{2}{5}a^{3}+\frac{1}{4}a^{2}-\frac{7}{20}a-\frac{1}{4}$, $\frac{1}{20}a^{10}-\frac{1}{20}a^{8}-\frac{1}{4}a^{7}+\frac{3}{20}a^{6}-\frac{1}{10}a^{4}+\frac{1}{4}a^{3}+\frac{7}{20}a^{2}-\frac{1}{4}a+\frac{1}{5}$, $\frac{1}{20}a^{11}-\frac{1}{20}a^{8}+\frac{1}{5}a^{7}-\frac{1}{20}a^{6}-\frac{7}{20}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{2}{5}a^{2}-\frac{3}{20}a+\frac{7}{20}$
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: | $a$, $\frac{3}{10}a^{10}+\frac{13}{10}a^{8}+\frac{7}{2}a^{6}-\frac{8}{5}a^{4}+\frac{49}{10}a^{2}+2$, $\frac{3}{5}a^{11}-\frac{7}{20}a^{10}+\frac{9}{4}a^{9}-\frac{5}{4}a^{8}+\frac{27}{5}a^{7}-\frac{29}{10}a^{6}-\frac{149}{20}a^{5}+\frac{47}{10}a^{4}+\frac{49}{4}a^{3}-8a^{2}+\frac{16}{5}a-\frac{9}{20}$, $\frac{17}{20}a^{11}+\frac{3}{5}a^{10}+\frac{63}{20}a^{9}+\frac{9}{4}a^{8}+\frac{151}{20}a^{7}+\frac{27}{5}a^{6}-\frac{107}{10}a^{5}-\frac{149}{20}a^{4}+\frac{359}{20}a^{3}+\frac{45}{4}a^{2}+\frac{22}{5}a+\frac{11}{5}$, $\frac{11}{10}a^{10}+\frac{41}{10}a^{8}+10a^{6}-\frac{66}{5}a^{4}+\frac{119}{5}a^{2}+\frac{7}{2}$ | 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: | \( 1500.58669489 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1500.58669489 \cdot 1}{2\cdot\sqrt{4452139149819904}}\cr\approx \mathstrut & 0.691872296335 \end{aligned}\]
Galois group
A non-solvable group of order 60 |
The 5 conjugacy class representatives for $A_5$ |
Character table for $A_5$ |
Intermediate fields
6.2.8340544.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.8340544.1 |
Degree 6 sibling: | 6.2.8340544.1 |
Degree 10 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Minimal sibling: | 5.1.8340544.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.59 | $x^{12} + 6 x^{11} + 22 x^{10} + 56 x^{9} + 126 x^{8} + 240 x^{7} + 332 x^{6} - 18 x^{5} - 459 x^{4} - 394 x^{3} - 344 x^{2} + 138 x + 423$ | $4$ | $3$ | $18$ | $A_4$ | $[2, 2]^{3}$ |
\(19\) | 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
19.10.8.1 | $x^{10} + 90 x^{9} + 3250 x^{8} + 59040 x^{7} + 544360 x^{6} + 2125046 x^{5} + 1090430 x^{4} + 296960 x^{3} + 1113560 x^{2} + 9728680 x + 34800945$ | $5$ | $2$ | $8$ | $D_5$ | $[\ ]_{5}^{2}$ |