Normalized defining polynomial
\( x^{10} - 3x^{9} - 3x^{8} + 26x^{7} - 20x^{6} - 54x^{5} + 100x^{4} - 34x^{3} - 87x^{2} + 147x - 71 \)
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: | \(26873856000000\) \(\medspace = 2^{18}\cdot 3^{8}\cdot 5^{6}\) | 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: | \(22.03\) | 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: | $2^{39/16}3^{4/3}5^{5/6}\approx 89.61842167211377$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{1}{3}a^{6}+\frac{1}{9}a^{5}-\frac{4}{9}a^{4}-\frac{2}{9}a^{3}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{2241}a^{9}-\frac{97}{2241}a^{8}+\frac{151}{2241}a^{7}-\frac{722}{2241}a^{6}+\frac{206}{747}a^{5}+\frac{40}{747}a^{4}+\frac{25}{2241}a^{3}-\frac{143}{2241}a^{2}-\frac{91}{2241}a-\frac{263}{2241}$
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: | $\frac{3}{83}a^{9}-\frac{46}{747}a^{8}-\frac{73}{747}a^{7}+\frac{142}{249}a^{6}-\frac{163}{747}a^{5}-\frac{329}{747}a^{4}+\frac{1505}{747}a^{3}-\frac{97}{83}a^{2}-\frac{1378}{747}a+\frac{1199}{747}$, $\frac{80}{747}a^{9}-\frac{23}{83}a^{8}-\frac{287}{747}a^{7}+\frac{1751}{747}a^{6}-\frac{526}{747}a^{5}-\frac{3929}{747}a^{4}+\frac{3577}{747}a^{3}+\frac{1259}{747}a^{2}-\frac{1154}{249}a+\frac{2947}{747}$, $\frac{41}{249}a^{9}-\frac{76}{249}a^{8}-\frac{200}{249}a^{7}+\frac{776}{249}a^{6}+\frac{106}{249}a^{5}-\frac{601}{83}a^{4}+\frac{1606}{249}a^{3}+\frac{362}{249}a^{2}-\frac{2569}{249}a+\frac{2248}{249}$, $\frac{2951}{2241}a^{9}-\frac{6122}{2241}a^{8}-\frac{14551}{2241}a^{7}+\frac{63317}{2241}a^{6}-\frac{152}{747}a^{5}-\frac{53272}{747}a^{4}+\frac{146981}{2241}a^{3}+\frac{35171}{2241}a^{2}-\frac{223721}{2241}a+\frac{224867}{2241}$, $\frac{65}{2241}a^{9}-\frac{578}{2241}a^{8}+\frac{1349}{2241}a^{7}+\frac{878}{2241}a^{6}-\frac{2629}{747}a^{5}+\frac{2932}{747}a^{4}+\frac{2870}{2241}a^{3}-\frac{16018}{2241}a^{2}+\frac{19981}{2241}a-\frac{9127}{2241}$ | 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: | \( 2163.79201504 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 2163.79201504 \cdot 1}{2\cdot\sqrt{26873856000000}}\cr\approx \mathstrut & 1.30106798378 \end{aligned}\]
Galois group
$C_2^4:A_5$ (as 10T34):
A non-solvable group of order 960 |
The 12 conjugacy class representatives for $C_2^4 : A_5$ |
Character table for $C_2^4 : A_5$ |
Intermediate fields
5.1.129600.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | 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 | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.8.18.79 | $x^{8} + 2 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 10$ | $8$ | $1$ | $18$ | $C_2\wr A_4$ | $[2, 2, 2, 2, 3]^{6}$ | |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ | |
3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ | |
\(5\) | 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |