Normalized defining polynomial
\( x^{10} - 30x^{7} - 45x^{6} + 12x^{5} + 100x^{4} + 300x^{3} + 45x^{2} - 270x + 36 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(6561000000000000\) \(\medspace = 2^{12}\cdot 3^{8}\cdot 5^{12}\) | 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: | \(38.17\) | 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: | $2^{15/8}3^{25/18}5^{271/200}\approx 149.3495805009198$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{5}+\frac{1}{5}$, $\frac{1}{5}a^{6}+\frac{1}{5}a$, $\frac{1}{25}a^{7}+\frac{2}{25}a^{6}+\frac{1}{25}a^{5}-\frac{1}{5}a^{3}+\frac{11}{25}a^{2}+\frac{12}{25}a-\frac{4}{25}$, $\frac{1}{75}a^{8}-\frac{1}{25}a^{6}+\frac{1}{25}a^{5}-\frac{3}{25}a^{3}-\frac{1}{15}a^{2}-\frac{11}{25}a+\frac{6}{25}$, $\frac{1}{2550}a^{9}-\frac{1}{255}a^{8}-\frac{6}{425}a^{7}-\frac{13}{425}a^{6}-\frac{61}{850}a^{5}+\frac{1}{425}a^{4}+\frac{106}{255}a^{3}-\frac{203}{1275}a^{2}-\frac{411}{850}a-\frac{13}{425}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{9}{170}a^{9}-\frac{16}{255}a^{8}+\frac{8}{85}a^{7}-\frac{147}{85}a^{6}-\frac{49}{170}a^{5}+\frac{27}{85}a^{4}+\frac{418}{85}a^{3}+\frac{2594}{255}a^{2}-\frac{1339}{170}a+\frac{23}{85}$, $\frac{11}{170}a^{9}-\frac{71}{425}a^{8}+\frac{81}{425}a^{7}-\frac{191}{85}a^{6}+\frac{2141}{850}a^{5}+\frac{169}{85}a^{4}+\frac{2579}{425}a^{3}+\frac{1976}{425}a^{2}-\frac{5233}{170}a+\frac{1493}{425}$, $\frac{77}{2550}a^{9}+\frac{2}{425}a^{8}-\frac{3}{425}a^{7}-\frac{389}{425}a^{6}-\frac{1297}{850}a^{5}+\frac{162}{425}a^{4}+\frac{4651}{1275}a^{3}+\frac{4372}{425}a^{2}+\frac{2217}{850}a-\frac{3806}{425}$, $\frac{77}{2550}a^{9}-\frac{28}{1275}a^{8}+\frac{14}{425}a^{7}-\frac{406}{425}a^{6}-\frac{481}{850}a^{5}+\frac{77}{425}a^{4}+\frac{3682}{1275}a^{3}+\frac{7217}{1275}a^{2}-\frac{1727}{850}a+\frac{87}{425}$, $\frac{97}{170}a^{9}-\frac{197}{255}a^{8}+\frac{484}{425}a^{7}-\frac{7967}{425}a^{6}-\frac{117}{850}a^{5}+\frac{342}{85}a^{4}+\frac{4286}{85}a^{3}+\frac{131012}{1275}a^{2}-\frac{89739}{850}a+\frac{3814}{425}$, $\frac{19}{510}a^{9}+\frac{307}{1275}a^{8}-\frac{128}{425}a^{7}-\frac{283}{425}a^{6}-\frac{8107}{850}a^{5}-\frac{32}{85}a^{4}+\frac{6847}{1275}a^{3}+\frac{41626}{1275}a^{2}+\frac{39869}{850}a-\frac{21771}{425}$, $\frac{203}{850}a^{9}-\frac{148}{425}a^{8}+\frac{222}{425}a^{7}-\frac{3361}{425}a^{6}+\frac{25}{34}a^{5}+\frac{609}{425}a^{4}+\frac{9007}{425}a^{3}+\frac{17067}{425}a^{2}-\frac{40587}{850}a+\frac{494}{85}$ (assuming GRH) | 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: | \( 40342.748639 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{2}\cdot 40342.748639 \cdot 1}{2\cdot\sqrt{6561000000000000}}\cr\approx \mathstrut & 0.62920212467 \end{aligned}\] (assuming GRH)
Galois group
$A_5^2:C_4$ (as 10T42):
A non-solvable group of order 14400 |
The 22 conjugacy class representatives for $A_5^2 : C_4$ |
Character table for $A_5^2 : C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
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 | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.8.12.24 | $x^{8} + 2 x^{7} + 2 x^{6} + 4 x^{5} + 16 x^{4} + 12 x^{3} - 12 x^{2} + 36$ | $4$ | $2$ | $12$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.6.2 | $x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |