Normalized defining polynomial
\( x^{20} - 144 x^{15} - 10889 x^{10} + 259506 x^{5} + 161051 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1315377880819141864776611328125=5^{31}\cdot 7^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.06$ | 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: | $5, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10} a^{10} + \frac{3}{10} a^{5} + \frac{1}{10}$, $\frac{1}{110} a^{11} - \frac{7}{110} a^{6} + \frac{21}{110} a$, $\frac{1}{110} a^{12} - \frac{7}{110} a^{7} + \frac{21}{110} a^{2}$, $\frac{1}{550} a^{13} + \frac{1}{275} a^{12} - \frac{1}{275} a^{11} - \frac{1}{50} a^{10} + \frac{103}{550} a^{8} + \frac{103}{275} a^{7} - \frac{103}{275} a^{6} - \frac{3}{50} a^{5} - \frac{199}{550} a^{3} + \frac{76}{275} a^{2} - \frac{76}{275} a - \frac{1}{50}$, $\frac{1}{550} a^{14} - \frac{1}{550} a^{12} - \frac{1}{275} a^{11} + \frac{1}{25} a^{10} + \frac{103}{550} a^{9} - \frac{103}{550} a^{7} - \frac{103}{275} a^{6} + \frac{3}{25} a^{5} - \frac{199}{550} a^{4} + \frac{199}{550} a^{2} - \frac{76}{275} a + \frac{1}{25}$, $\frac{1}{1404490450} a^{15} + \frac{9162221}{702245225} a^{10} - \frac{215998866}{702245225} a^{5} - \frac{44521051}{127680950}$, $\frac{1}{1404490450} a^{16} + \frac{5556347}{1404490450} a^{11} - \frac{342621067}{1404490450} a^{6} + \frac{323314447}{702245225} a$, $\frac{1}{15449394950} a^{17} - \frac{29142064}{7724697475} a^{12} + \frac{1456621579}{7724697475} a^{7} - \frac{2098511531}{15449394950} a^{2}$, $\frac{1}{169943344450} a^{18} - \frac{86373937}{169943344450} a^{13} - \frac{1}{275} a^{12} + \frac{1}{275} a^{11} + \frac{1}{50} a^{10} + \frac{46368177681}{169943344450} a^{8} - \frac{103}{275} a^{7} + \frac{103}{275} a^{6} + \frac{3}{50} a^{5} + \frac{1894075541}{16994334445} a^{3} - \frac{76}{275} a^{2} + \frac{76}{275} a + \frac{1}{50}$, $\frac{1}{1869376788950} a^{19} + \frac{729282779}{934688394475} a^{14} + \frac{1}{550} a^{12} + \frac{1}{275} a^{11} - \frac{1}{25} a^{10} + \frac{442635161733}{934688394475} a^{9} + \frac{103}{550} a^{7} + \frac{103}{275} a^{6} - \frac{3}{25} a^{5} + \frac{112242903631}{373875357790} a^{4} - \frac{199}{550} a^{2} + \frac{76}{275} a - \frac{1}{25}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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: | \( 19025685.222571 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times F_5$ (as 20T9):
| A solvable group of order 40 |
| The 10 conjugacy class representatives for $C_2\times F_5$ |
| Character table for $C_2\times F_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.6125.1, 5.1.78125.1, 10.2.30517578125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 10 siblings: | data not computed |
| Degree 20 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 | R | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 | ||||||
| $7$ | 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |