Normalized defining polynomial
\( x^{10} - 4x^{9} + 9x^{8} - 9x^{7} - 2x^{6} + 12x^{5} - 8x^{4} - 12x^{3} + 27x^{2} - 14x + 5 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-62910546875\) \(\medspace = -\,5^{8}\cdot 11^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.02\) | 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))
| |
Galois root discriminant: | $5^{8/5}11^{1/2}\approx 43.55603614321637$ | ||
Ramified primes: | \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $5$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{35}a^{8}+\frac{17}{35}a^{7}-\frac{8}{35}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{17}{35}a^{3}-\frac{11}{35}a^{2}+\frac{2}{5}a-\frac{2}{7}$, $\frac{1}{1225}a^{9}-\frac{6}{1225}a^{8}-\frac{22}{175}a^{7}-\frac{226}{1225}a^{6}-\frac{24}{49}a^{5}-\frac{538}{1225}a^{4}-\frac{332}{1225}a^{3}+\frac{127}{1225}a^{2}+\frac{473}{1225}a+\frac{53}{245}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{86}{1225}a^{9}-\frac{271}{1225}a^{8}+\frac{103}{175}a^{7}-\frac{571}{1225}a^{6}-\frac{6}{49}a^{5}+\frac{1507}{1225}a^{4}-\frac{1112}{1225}a^{3}-\frac{1573}{1225}a^{2}+\frac{2458}{1225}a-\frac{97}{245}$, $\frac{94}{1225}a^{9}-\frac{389}{1225}a^{8}+\frac{107}{175}a^{7}-\frac{594}{1225}a^{6}-\frac{30}{49}a^{5}+\frac{1053}{1225}a^{4}-\frac{58}{1225}a^{3}-\frac{2237}{1225}a^{2}+\frac{1587}{1225}a-\frac{23}{245}$, $\frac{491}{1225}a^{9}-\frac{1686}{1225}a^{8}+\frac{69}{25}a^{7}-\frac{2221}{1225}a^{6}-\frac{108}{49}a^{5}+\frac{4642}{1225}a^{4}-\frac{717}{1225}a^{3}-\frac{6628}{1225}a^{2}+\frac{9783}{1225}a-\frac{262}{245}$, $\frac{94}{1225}a^{9}-\frac{389}{1225}a^{8}+\frac{107}{175}a^{7}-\frac{594}{1225}a^{6}-\frac{30}{49}a^{5}+\frac{1053}{1225}a^{4}-\frac{58}{1225}a^{3}-\frac{2237}{1225}a^{2}+\frac{2812}{1225}a-\frac{23}{245}$ | 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: | \( 29.8937495286 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 29.8937495286 \cdot 1}{2\cdot\sqrt{62910546875}}\cr\approx \mathstrut & 0.583563348016 \end{aligned}\]
Galois group
$C_5\times D_5$ (as 10T6):
A solvable group of order 50 |
The 20 conjugacy class representatives for $D_5\times C_5$ |
Character table for $D_5\times C_5$ |
Intermediate fields
\(\Q(\sqrt{-11}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 25 sibling: | 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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
5.5.8.2 | $x^{5} + 20 x^{4} + 5$ | $5$ | $1$ | $8$ | $C_5$ | $[2]$ | |
\(11\) | 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.25.5t1.a.c | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ | |
1.25.5t1.a.b | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ | |
1.275.10t1.a.a | $1$ | $ 5^{2} \cdot 11 $ | 10.0.24574432373046875.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
1.25.5t1.a.a | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ | |
1.275.10t1.a.d | $1$ | $ 5^{2} \cdot 11 $ | 10.0.24574432373046875.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
1.275.10t1.a.b | $1$ | $ 5^{2} \cdot 11 $ | 10.0.24574432373046875.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
1.25.5t1.a.d | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ | |
1.275.10t1.a.c | $1$ | $ 5^{2} \cdot 11 $ | 10.0.24574432373046875.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
* | 2.275.10t6.a.c | $2$ | $ 5^{2} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |
2.6875.5t2.a.a | $2$ | $ 5^{4} \cdot 11 $ | 5.1.47265625.1 | $D_{5}$ (as 5T2) | $1$ | $0$ | |
2.6875.10t6.a.a | $2$ | $ 5^{4} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ | |
2.6875.10t6.a.c | $2$ | $ 5^{4} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ | |
2.6875.10t6.a.b | $2$ | $ 5^{4} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ | |
* | 2.275.10t6.a.b | $2$ | $ 5^{2} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |
* | 2.275.10t6.a.d | $2$ | $ 5^{2} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |
2.6875.5t2.a.b | $2$ | $ 5^{4} \cdot 11 $ | 5.1.47265625.1 | $D_{5}$ (as 5T2) | $1$ | $0$ | |
* | 2.275.10t6.a.a | $2$ | $ 5^{2} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |
2.6875.10t6.a.d | $2$ | $ 5^{4} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |