Normalized defining polynomial
\( x^{12} - 5 x^{11} + 16 x^{10} - 35 x^{9} + 61 x^{8} - 85 x^{7} + 106 x^{6} - 120 x^{5} + 126 x^{4} + \cdots + 5 \)
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: | \(1803751953125\) \(\medspace = 5^{9}\cdot 31^{4}\) | 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: | \(10.50\) | 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: | $5^{3/4}31^{2/3}\approx 32.99655748259533$ | ||
Ramified primes: | \(5\), \(31\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{60}a^{10}+\frac{2}{15}a^{9}+\frac{3}{20}a^{8}-\frac{1}{10}a^{7}+\frac{1}{15}a^{6}+\frac{13}{60}a^{5}+\frac{11}{60}a^{4}-\frac{1}{3}a^{3}+\frac{1}{12}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{60}a^{11}+\frac{1}{12}a^{9}+\frac{1}{30}a^{8}-\frac{7}{15}a^{7}-\frac{19}{60}a^{6}+\frac{9}{20}a^{5}-\frac{7}{15}a^{4}+\frac{1}{12}a^{3}+\frac{1}{4}a^{2}-\frac{5}{12}a-\frac{1}{3}$
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: | \( \frac{3}{10} a^{11} - \frac{79}{60} a^{10} + \frac{119}{30} a^{9} - \frac{95}{12} a^{8} + \frac{79}{6} a^{7} - \frac{509}{30} a^{6} + \frac{1259}{60} a^{5} - \frac{451}{20} a^{4} + \frac{139}{6} a^{3} - \frac{209}{12} a^{2} + \frac{133}{12} a - \frac{25}{12} \) (order $10$) | 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: | $\frac{3}{10}a^{11}-\frac{79}{60}a^{10}+\frac{119}{30}a^{9}-\frac{95}{12}a^{8}+\frac{79}{6}a^{7}-\frac{509}{30}a^{6}+\frac{1259}{60}a^{5}-\frac{451}{20}a^{4}+\frac{139}{6}a^{3}-\frac{209}{12}a^{2}+\frac{133}{12}a-\frac{37}{12}$, $\frac{4}{15}a^{11}-\frac{13}{12}a^{10}+\frac{10}{3}a^{9}-\frac{413}{60}a^{8}+\frac{361}{30}a^{7}-\frac{82}{5}a^{6}+\frac{1247}{60}a^{5}-\frac{1363}{60}a^{4}+\frac{70}{3}a^{3}-\frac{79}{4}a^{2}+\frac{173}{12}a-\frac{55}{12}$, $\frac{7}{30}a^{11}-\frac{61}{60}a^{10}+\frac{91}{30}a^{9}-\frac{361}{60}a^{8}+\frac{99}{10}a^{7}-\frac{25}{2}a^{6}+\frac{181}{12}a^{5}-\frac{321}{20}a^{4}+\frac{97}{6}a^{3}-\frac{49}{4}a^{2}+\frac{91}{12}a-\frac{19}{12}$, $\frac{3}{5}a^{11}-\frac{161}{60}a^{10}+\frac{41}{5}a^{9}-\frac{339}{20}a^{8}+\frac{859}{30}a^{7}-\frac{572}{15}a^{6}+\frac{2819}{60}a^{5}-\frac{3119}{60}a^{4}+\frac{161}{3}a^{3}-\frac{533}{12}a^{2}+\frac{115}{4}a-\frac{95}{12}$, $\frac{1}{10}a^{11}-\frac{7}{30}a^{10}+\frac{19}{30}a^{9}-\frac{9}{10}a^{8}+\frac{8}{5}a^{7}-\frac{11}{6}a^{6}+\frac{8}{3}a^{5}-\frac{71}{30}a^{4}+\frac{13}{6}a^{3}-\frac{5}{3}a^{2}+2a-\frac{1}{2}$ | 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: | \( 75.4332875798 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 75.4332875798 \cdot 1}{10\cdot\sqrt{1803751953125}}\cr\approx \mathstrut & 0.345584168433 \end{aligned}\]
Galois group
$C_3^2:C_4$ (as 12T17):
A solvable group of order 36 |
The 6 conjugacy class representatives for $(C_3\times C_3):C_4$ |
Character table for $(C_3\times C_3):C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 6.2.600625.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 36 |
Degree 6 siblings: | 6.2.600625.1, 6.2.577200625.1 |
Degree 9 sibling: | 9.1.13867245015625.1 |
Degree 12 sibling: | deg 12 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 6.2.600625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{3}$ | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(31\) | 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |