Normalized defining polynomial
\( x^{10} - 3x^{9} + 4x^{8} + 2x^{7} - 19x^{6} + 35x^{5} - 40x^{4} + 40x^{3} - 45x^{2} + 49x - 22 \)
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: | \(575073488896\) \(\medspace = 2^{12}\cdot 17^{4}\cdot 41^{2}\) | 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: | \(15.00\) | 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: | $2^{3/2}17^{2/3}41^{1/2}\approx 119.73915879740562$ | ||
Ramified primes: | \(2\), \(17\), \(41\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{106543}a^{9}-\frac{14940}{106543}a^{8}-\frac{48801}{106543}a^{7}-\frac{26667}{106543}a^{6}-\frac{39317}{106543}a^{5}+\frac{13048}{106543}a^{4}-\frac{30869}{106543}a^{3}-\frac{27811}{106543}a^{2}+\frac{1705}{106543}a-\frac{3759}{106543}$
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: | $\frac{16155}{106543}a^{9}-\frac{35805}{106543}a^{8}+\frac{38045}{106543}a^{7}+\frac{54507}{106543}a^{6}-\frac{276398}{106543}a^{5}+\frac{368015}{106543}a^{4}-\frac{387084}{106543}a^{3}+\frac{324755}{106543}a^{2}-\frac{476534}{106543}a+\frac{429037}{106543}$, $\frac{1969}{106543}a^{9}-\frac{10992}{106543}a^{8}+\frac{12617}{106543}a^{7}+\frac{18376}{106543}a^{6}-\frac{64955}{106543}a^{5}+\frac{121192}{106543}a^{4}-\frac{51551}{106543}a^{3}+\frac{3243}{106543}a^{2}-\frac{52231}{106543}a+\frac{56539}{106543}$, $\frac{9815}{106543}a^{9}-\frac{32932}{106543}a^{8}+\frac{35513}{106543}a^{7}+\frac{39546}{106543}a^{6}-\frac{210695}{106543}a^{5}+\frac{321063}{106543}a^{4}-\frac{290572}{106543}a^{3}+\frac{317830}{106543}a^{2}-\frac{418848}{106543}a+\frac{395465}{106543}$, $\frac{11149}{106543}a^{9}-\frac{39351}{106543}a^{8}+\frac{32752}{106543}a^{7}+\frac{51130}{106543}a^{6}-\frac{240417}{106543}a^{5}+\frac{360586}{106543}a^{4}-\frac{237677}{106543}a^{3}+\frac{294920}{106543}a^{2}-\frac{488324}{106543}a+\frac{281937}{106543}$, $\frac{7027}{106543}a^{9}-\frac{38525}{106543}a^{8}+\frac{37290}{106543}a^{7}+\frac{20128}{106543}a^{6}-\frac{227646}{106543}a^{5}+\frac{380945}{106543}a^{4}-\frac{314544}{106543}a^{3}+\frac{291594}{106543}a^{2}-\frac{377953}{106543}a+\frac{434343}{106543}$ | 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: | \( 254.537119467 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 254.537119467 \cdot 1}{2\cdot\sqrt{575073488896}}\cr\approx \mathstrut & 1.04625831048 \end{aligned}\]
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
5.1.18496.1 |
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 | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
2.8.12.16 | $x^{8} + 8 x^{7} + 32 x^{6} + 78 x^{5} + 137 x^{4} + 186 x^{3} + 128 x^{2} - 10 x + 7$ | $4$ | $2$ | $12$ | $A_4\times C_2$ | $[2, 2]^{6}$ | |
\(17\) | 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
17.6.4.1 | $x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
41.4.2.2 | $x^{4} - 45592 x^{3} - 53825497 x^{2} - 1482642 x + 10086$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |