Normalized defining polynomial
\( x^{20} - 36 x^{15} + 2486 x^{10} - 47916 x^{5} + 1771561 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(23383113568432629108428955078125=5^{27}\cdot 11^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.02$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{1}{10} a^{2} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{4} - \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{660} a^{10} + \frac{27}{110} a^{5} + \frac{11}{60}$, $\frac{1}{660} a^{11} + \frac{27}{110} a^{6} + \frac{11}{60} a$, $\frac{1}{660} a^{12} + \frac{1}{22} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{60} a^{2} + \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{7260} a^{13} - \frac{3}{605} a^{8} - \frac{1}{10} a^{6} + \frac{1}{10} a^{5} - \frac{247}{660} a^{3} - \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{36300} a^{14} - \frac{1}{36300} a^{13} + \frac{1}{3300} a^{12} - \frac{1}{3300} a^{11} + \frac{1}{3300} a^{10} - \frac{124}{3025} a^{9} + \frac{124}{3025} a^{8} + \frac{27}{550} a^{7} - \frac{27}{550} a^{6} + \frac{27}{550} a^{5} + \frac{1601}{3300} a^{4} - \frac{1601}{3300} a^{3} - \frac{49}{300} a^{2} + \frac{49}{300} a - \frac{49}{300}$, $\frac{1}{798600} a^{15} - \frac{157}{798600} a^{10} - \frac{10147}{72600} a^{5} - \frac{89}{600}$, $\frac{1}{798600} a^{16} - \frac{157}{798600} a^{11} - \frac{10147}{72600} a^{6} - \frac{89}{600} a$, $\frac{1}{8784600} a^{17} - \frac{1367}{8784600} a^{12} - \frac{20707}{798600} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{1939}{6600} a^{2} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{8784600} a^{18} - \frac{157}{8784600} a^{13} - \frac{24667}{798600} a^{8} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} + \frac{2191}{6600} a^{3} + \frac{1}{10} a - \frac{1}{10}$, $\frac{1}{96630600} a^{19} + \frac{259}{19326120} a^{14} - \frac{1}{36300} a^{13} + \frac{1}{3300} a^{12} - \frac{1}{3300} a^{11} + \frac{1}{3300} a^{10} + \frac{98357}{8784600} a^{9} + \frac{124}{3025} a^{8} + \frac{27}{550} a^{7} - \frac{137}{550} a^{6} + \frac{41}{275} a^{5} + \frac{30643}{72600} a^{4} - \frac{1601}{3300} a^{3} - \frac{49}{300} a^{2} - \frac{11}{300} a - \frac{19}{300}$
Class group and class number
$C_{5}$, which has order $5$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{7}{798600} a^{15} - \frac{1099}{798600} a^{10} + \frac{1571}{72600} a^{5} - \frac{623}{600} \) (order $10$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 45537996.331 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 100 |
| The 13 conjugacy class representatives for $D_5.D_5$ |
| Character table for $D_5.D_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 25 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | Data not computed | ||||||
| $11$ | 11.5.4.1 | $x^{5} + 297$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.2 | $x^{5} - 891$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.5 | $x^{5} - 99$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.0.1 | $x^{5} + x^{2} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |