Normalized defining polynomial
\( x^{10} - x^{9} - 13x^{8} + 8x^{7} + 46x^{6} - 11x^{5} - 52x^{4} + 7x^{3} + 18x^{2} - 3x - 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(669871503125\) \(\medspace = 5^{5}\cdot 11^{8}\) | 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: | \(15.23\) | 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: | $5^{1/2}11^{4/5}\approx 15.226467164777734$ | ||
Ramified primes: | \(5\), \(11\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(55=5\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{55}(1,·)$, $\chi_{55}(34,·)$, $\chi_{55}(4,·)$, $\chi_{55}(9,·)$, $\chi_{55}(14,·)$, $\chi_{55}(16,·)$, $\chi_{55}(49,·)$, $\chi_{55}(36,·)$, $\chi_{55}(26,·)$, $\chi_{55}(31,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{331}a^{9}+\frac{100}{331}a^{8}+\frac{157}{331}a^{7}-\frac{23}{331}a^{6}+\frac{40}{331}a^{5}+\frac{57}{331}a^{4}+\frac{78}{331}a^{3}-\frac{59}{331}a^{2}+\frac{17}{331}a+\frac{59}{331}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{763}{331}a^{9}-\frac{492}{331}a^{8}-\frac{9961}{331}a^{7}+\frac{2311}{331}a^{6}+\frac{34492}{331}a^{5}+\frac{6088}{331}a^{4}-\frac{34159}{331}a^{3}-\frac{9931}{331}a^{2}+\frac{8006}{331}a+\frac{1325}{331}$, $\frac{229}{331}a^{9}-\frac{270}{331}a^{8}-\frac{2774}{331}a^{7}+\frac{2015}{331}a^{6}+\frac{8498}{331}a^{5}-\frac{1180}{331}a^{4}-\frac{7625}{331}a^{3}-\frac{1926}{331}a^{2}+\frac{1576}{331}a+\frac{933}{331}$, $\frac{271}{331}a^{9}-\frac{42}{331}a^{8}-\frac{3793}{331}a^{7}-\frac{606}{331}a^{6}+\frac{14481}{331}a^{5}+\frac{5517}{331}a^{4}-\frac{15272}{331}a^{3}-\frac{5728}{331}a^{2}+\frac{3283}{331}a+\frac{763}{331}$, $\frac{159}{331}a^{9}+\frac{12}{331}a^{8}-\frac{2179}{331}a^{7}-\frac{1009}{331}a^{6}+\frac{8015}{331}a^{5}+\frac{6415}{331}a^{4}-\frac{8451}{331}a^{3}-\frac{7064}{331}a^{2}+\frac{2703}{331}a+\frac{1106}{331}$, $\frac{430}{331}a^{9}-\frac{30}{331}a^{8}-\frac{5972}{331}a^{7}-\frac{1615}{331}a^{6}+\frac{22496}{331}a^{5}+\frac{11932}{331}a^{4}-\frac{23723}{331}a^{3}-\frac{12792}{331}a^{2}+\frac{5986}{331}a+\frac{1869}{331}$, $\frac{62}{331}a^{9}+\frac{242}{331}a^{8}-\frac{1189}{331}a^{7}-\frac{3081}{331}a^{6}+\frac{5790}{331}a^{5}+\frac{9492}{331}a^{4}-\frac{5756}{331}a^{3}-\frac{7299}{331}a^{2}+\frac{1054}{331}a+\frac{1010}{331}$, $\frac{28}{331}a^{9}+\frac{152}{331}a^{8}-\frac{569}{331}a^{7}-\frac{1968}{331}a^{6}+\frac{2775}{331}a^{5}+\frac{6561}{331}a^{4}-\frac{2119}{331}a^{3}-\frac{6617}{331}a^{2}-\frac{517}{331}a+\frac{1652}{331}$, $\frac{1193}{331}a^{9}-\frac{522}{331}a^{8}-\frac{15933}{331}a^{7}+\frac{696}{331}a^{6}+\frac{56988}{331}a^{5}+\frac{18020}{331}a^{4}-\frac{57882}{331}a^{3}-\frac{22723}{331}a^{2}+\frac{13992}{331}a+\frac{3194}{331}$, $\frac{492}{331}a^{9}-\frac{450}{331}a^{8}-\frac{6168}{331}a^{7}+\frac{2917}{331}a^{6}+\frac{20011}{331}a^{5}+\frac{571}{331}a^{4}-\frac{18887}{331}a^{3}-\frac{4203}{331}a^{2}+\frac{4723}{331}a+\frac{562}{331}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 274.696482776 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 274.696482776 \cdot 1}{2\cdot\sqrt{669871503125}}\cr\approx \mathstrut & 0.171841204528 \end{aligned}\]
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.10.0.1}{10} }$ | R | ${\href{/padicField/7.10.0.1}{10} }$ | R | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.55.10t1.a.a | $1$ | $ 5 \cdot 11 $ | 10.10.669871503125.1 | $C_{10}$ (as 10T1) | $0$ | $1$ |
* | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.55.10t1.a.b | $1$ | $ 5 \cdot 11 $ | 10.10.669871503125.1 | $C_{10}$ (as 10T1) | $0$ | $1$ |
* | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.55.10t1.a.c | $1$ | $ 5 \cdot 11 $ | 10.10.669871503125.1 | $C_{10}$ (as 10T1) | $0$ | $1$ |
* | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.55.10t1.a.d | $1$ | $ 5 \cdot 11 $ | 10.10.669871503125.1 | $C_{10}$ (as 10T1) | $0$ | $1$ |