Normalized defining polynomial
\( x^{10} - 2x^{9} - 3x^{8} + 12x^{7} - 7x^{6} - 20x^{5} + 17x^{4} + 16x^{3} - 6x^{2} - 6x - 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-155838906487\) \(\medspace = -\,11^{8}\cdot 727\) | 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: | \(13.16\) | 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: | $11^{4/5}727^{1/2}\approx 183.60366814782304$ | ||
Ramified primes: | \(11\), \(727\) | 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: | \(\Q(\sqrt{-727}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{5}{2}a^{9}-\frac{13}{2}a^{8}-4a^{7}+33a^{6}-36a^{5}-\frac{65}{2}a^{4}+\frac{127}{2}a^{3}+9a^{2}-24a-\frac{11}{2}$, $a$, $2a^{9}-4a^{8}-6a^{7}+\frac{49}{2}a^{6}-15a^{5}-41a^{4}+39a^{3}+\frac{55}{2}a^{2}-\frac{35}{2}a-\frac{15}{2}$, $2a^{9}-\frac{11}{2}a^{8}-2a^{7}+26a^{6}-\frac{67}{2}a^{5}-17a^{4}+50a^{3}-5a^{2}-13a-\frac{3}{2}$, $\frac{5}{2}a^{9}-\frac{13}{2}a^{8}-\frac{9}{2}a^{7}+34a^{6}-35a^{5}-\frac{75}{2}a^{4}+68a^{3}+\frac{29}{2}a^{2}-\frac{57}{2}a-\frac{17}{2}$, $4a^{9}-\frac{19}{2}a^{8}-8a^{7}+\frac{101}{2}a^{6}-\frac{97}{2}a^{5}-58a^{4}+89a^{3}+\frac{45}{2}a^{2}-\frac{61}{2}a-8$, $\frac{1}{2}a^{9}-\frac{5}{2}a^{8}+\frac{3}{2}a^{7}+\frac{19}{2}a^{6}-20a^{5}+\frac{7}{2}a^{4}+29a^{3}-13a^{2}-11a-1$, $\frac{1}{2}a^{9}-a^{8}-2a^{7}+7a^{6}-\frac{5}{2}a^{5}-\frac{31}{2}a^{4}+\frac{27}{2}a^{3}+14a^{2}-10a-4$ | 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: | \( 84.3760276432 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{1}\cdot 84.3760276432 \cdot 1}{2\cdot\sqrt{155838906487}}\cr\approx \mathstrut & 0.171898056777 \end{aligned}\]
Galois group
$C_2\wr C_5$ (as 10T14):
A solvable group of order 160 |
The 16 conjugacy class representatives for $C_2 \times (C_2^4 : C_5)$ |
Character table for $C_2 \times (C_2^4 : C_5)$ |
Intermediate fields
\(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 32 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 | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
\(727\) | $\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{727}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |