Normalized defining polynomial
\( x^{10} - 10x^{7} - 5x^{6} + 19x^{5} + 25x^{4} - 20x^{3} - 10x - 7 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(7476806640625\) \(\medspace = 5^{16}\cdot 7^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.38\) | 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: | $5^{8/5}7^{1/2}\approx 34.74569691024434$ | ||
Ramified primes: | \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{268051}a^{9}+\frac{63723}{268051}a^{8}-\frac{83870}{268051}a^{7}-\frac{47182}{268051}a^{6}-\frac{118575}{268051}a^{5}-\frac{133118}{268051}a^{4}+\frac{63657}{268051}a^{3}-\frac{792}{268051}a^{2}-\frac{75028}{268051}a-\frac{7374}{38293}$
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)
| |
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: | $\frac{14075}{268051}a^{9}+\frac{2579}{268051}a^{8}+\frac{26354}{268051}a^{7}-\frac{124323}{268051}a^{6}-\frac{57599}{268051}a^{5}+\frac{40640}{268051}a^{4}+\frac{145833}{268051}a^{3}-\frac{157309}{268051}a^{2}+\frac{101840}{268051}a-\frac{15020}{38293}$, $\frac{2579}{268051}a^{9}+\frac{26354}{268051}a^{8}+\frac{16427}{268051}a^{7}+\frac{12776}{268051}a^{6}-\frac{226785}{268051}a^{5}-\frac{206042}{268051}a^{4}+\frac{124191}{268051}a^{3}+\frac{369891}{268051}a^{2}+\frac{35610}{268051}a+\frac{90661}{38293}$, $\frac{19736}{268051}a^{9}-\frac{58164}{268051}a^{8}-\frac{43395}{268051}a^{7}-\frac{242829}{268051}a^{6}+\frac{425132}{268051}a^{5}+\frac{1023207}{268051}a^{4}-\frac{20485}{268051}a^{3}-\frac{1960311}{268051}a^{2}+\frac{229167}{268051}a-\frac{58157}{38293}$, $\frac{11803}{268051}a^{9}-\frac{28537}{268051}a^{8}-\frac{5267}{268051}a^{7}-\frac{147219}{268051}a^{6}+\frac{221597}{268051}a^{5}+\frac{391259}{268051}a^{4}-\frac{3382}{268051}a^{3}-\frac{770344}{268051}a^{2}+\frac{85020}{268051}a-\frac{33626}{38293}$, $\frac{89396}{268051}a^{9}-\frac{38544}{268051}a^{8}+\frac{12001}{268051}a^{7}-\frac{903740}{268051}a^{6}-\frac{53905}{268051}a^{5}+\frac{1783774}{268051}a^{4}+\frac{1566748}{268051}a^{3}-\frac{2448627}{268051}a^{2}+\frac{773187}{268051}a-\frac{183574}{38293}$ | 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: | \( 541.927062633 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 541.927062633 \cdot 1}{2\cdot\sqrt{7476806640625}}\cr\approx \mathstrut & 0.617777708796 \end{aligned}\]
Galois group
$C_2^4:C_5$ (as 10T8):
A solvable group of order 80 |
The 8 conjugacy class representatives for $C_2^4 : C_5$ |
Character table for $C_2^4 : C_5$ |
Intermediate fields
5.5.390625.1 |
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 16 sibling: | data not computed |
Degree 20 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 | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | R | R | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.5.8.2 | $x^{5} + 20 x^{4} + 5$ | $5$ | $1$ | $8$ | $C_5$ | $[2]$ |
5.5.8.2 | $x^{5} + 20 x^{4} + 5$ | $5$ | $1$ | $8$ | $C_5$ | $[2]$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |