Normalized defining polynomial
\( x^{10} + 20x^{8} - 10x^{7} + 130x^{6} - 140x^{5} + 295x^{4} - 520x^{3} + 200x^{2} - 20 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(14762250000000000\) \(\medspace = 2^{10}\cdot 3^{10}\cdot 5^{12}\) | 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: | \(41.39\) | 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: | $2^{2}3^{11/6}5^{271/200}\approx 265.39127904484207$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{36}a^{7}-\frac{1}{12}a^{6}+\frac{17}{36}a^{5}+\frac{1}{36}a^{4}-\frac{11}{36}a^{3}-\frac{1}{36}a^{2}-\frac{1}{3}a-\frac{1}{18}$, $\frac{1}{36}a^{8}-\frac{1}{9}a^{6}-\frac{2}{9}a^{5}-\frac{2}{9}a^{4}-\frac{5}{18}a^{3}-\frac{5}{12}a^{2}+\frac{5}{18}a-\frac{1}{2}$, $\frac{1}{15768}a^{9}+\frac{4}{1971}a^{8}+\frac{7}{657}a^{7}+\frac{137}{3942}a^{6}-\frac{11}{27}a^{5}-\frac{1019}{7884}a^{4}+\frac{341}{15768}a^{3}-\frac{239}{2628}a^{2}+\frac{1903}{7884}a-\frac{982}{1971}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{7}{584}a^{9}-\frac{7}{1314}a^{8}+\frac{155}{657}a^{7}-\frac{275}{1314}a^{6}+\frac{14}{9}a^{5}-\frac{1835}{876}a^{4}+\frac{20315}{5256}a^{3}-\frac{17285}{2628}a^{2}+\frac{9805}{2628}a+\frac{476}{657}$, $\frac{821}{15768}a^{9}+\frac{217}{3942}a^{8}+\frac{1493}{1314}a^{7}+\frac{1598}{1971}a^{6}+\frac{433}{54}a^{5}+\frac{24947}{7884}a^{4}+\frac{277333}{15768}a^{3}+\frac{33}{292}a^{2}-\frac{36337}{7884}a-\frac{1616}{1971}$, $\frac{21763}{15768}a^{9}+\frac{5473}{7884}a^{8}+\frac{36553}{1314}a^{7}+\frac{142}{1971}a^{6}+\frac{9545}{54}a^{5}-\frac{828815}{7884}a^{4}+\frac{5324039}{15768}a^{3}-\frac{39571}{73}a^{2}-\frac{202457}{7884}a+\frac{119929}{3942}$, $\frac{325}{15768}a^{9}-\frac{247}{3942}a^{8}+\frac{77}{219}a^{7}-\frac{3217}{3942}a^{6}+\frac{49}{27}a^{5}-\frac{18443}{7884}a^{4}+\frac{2201}{15768}a^{3}+\frac{1019}{2628}a^{2}-\frac{857}{7884}a-\frac{67}{1971}$, $\frac{907}{3942}a^{9}-\frac{599}{1971}a^{8}+\frac{12889}{2628}a^{7}-\frac{68297}{7884}a^{6}+\frac{4231}{108}a^{5}-\frac{617573}{7884}a^{4}+\frac{1203961}{7884}a^{3}-\frac{673277}{2628}a^{2}+\frac{582841}{1971}a-\frac{641459}{3942}$ (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: | \( 38224.7574432 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 38224.7574432 \cdot 1}{2\cdot\sqrt{14762250000000000}}\cr\approx \mathstrut & 0.980658799022 \end{aligned}\] (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 | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.6.4 | $x^{4} + 4 x^{3} + 24 x^{2} + 88 x + 124$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.8.1 | $x^{6} - 6 x^{5} + 45 x^{4} + 6 x^{3} - 18 x^{2} + 9$ | $3$ | $2$ | $8$ | $C_3^2:C_4$ | $[2, 2]^{4}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |