Normalized defining polynomial
\( x^{40} - x^{35} + x^{25} - x^{20} + x^{15} - x^{5} + 1 \)
Invariants
Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(29534212676193502024324377686070874915458261966705322265625\)
\(\medspace = 3^{20}\cdot 5^{70}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(28.96\) | 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: | $3^{1/2}5^{7/4}\approx 28.957304632206725$ | ||
Ramified primes: |
\(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(75=3\cdot 5^{2}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{75}(1,·)$, $\chi_{75}(2,·)$, $\chi_{75}(4,·)$, $\chi_{75}(7,·)$, $\chi_{75}(8,·)$, $\chi_{75}(11,·)$, $\chi_{75}(13,·)$, $\chi_{75}(14,·)$, $\chi_{75}(16,·)$, $\chi_{75}(17,·)$, $\chi_{75}(19,·)$, $\chi_{75}(22,·)$, $\chi_{75}(23,·)$, $\chi_{75}(26,·)$, $\chi_{75}(28,·)$, $\chi_{75}(29,·)$, $\chi_{75}(31,·)$, $\chi_{75}(32,·)$, $\chi_{75}(34,·)$, $\chi_{75}(37,·)$, $\chi_{75}(38,·)$, $\chi_{75}(41,·)$, $\chi_{75}(43,·)$, $\chi_{75}(44,·)$, $\chi_{75}(46,·)$, $\chi_{75}(47,·)$, $\chi_{75}(49,·)$, $\chi_{75}(52,·)$, $\chi_{75}(53,·)$, $\chi_{75}(56,·)$, $\chi_{75}(58,·)$, $\chi_{75}(59,·)$, $\chi_{75}(61,·)$, $\chi_{75}(62,·)$, $\chi_{75}(64,·)$, $\chi_{75}(67,·)$, $\chi_{75}(68,·)$, $\chi_{75}(71,·)$, $\chi_{75}(73,·)$, $\chi_{75}(74,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{11}$, which has order $11$ (assuming GRH)
Unit group
Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: |
\( -a \)
(order $150$)
| 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: |
$a^{39}-a^{29}+a^{24}-a^{4}$, $a^{10}-1$, $a^{23}-a^{18}+a^{8}$, $a^{3}+1$, $a^{12}+1$, $a^{6}+1$, $a^{27}-a^{26}-a$, $a^{33}-a^{29}-a^{4}$, $a^{18}+1$, $a^{38}-a^{33}+a^{21}-a^{18}+a^{13}-a^{3}$, $a^{39}+a^{3}$, $a^{2}-1$, $a-1$, $a^{4}-1$, $a^{8}-1$, $a^{13}-1$, $a^{35}+a^{34}-a^{30}+a^{20}-a^{15}+a^{10}+a^{9}-1$, $a^{16}-1$, $a^{31}+a^{12}+a^{6}$
| 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: | \( 32146857615014.19 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 32146857615014.19 \cdot 11}{150\cdot\sqrt{29534212676193502024324377686070874915458261966705322265625}}\cr\approx \mathstrut & 0.126146330271225 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{20}$ (as 40T2):
An abelian group of order 40 |
The 40 conjugacy class representatives for $C_2\times C_{20}$ |
Character table for $C_2\times C_{20}$ |
Intermediate fields
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 | $20^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{10}$ | ${\href{/padicField/11.10.0.1}{10} }^{4}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{8}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\)
| Deg $40$ | $2$ | $20$ | $20$ | |||
\(5\)
| Deg $40$ | $20$ | $2$ | $70$ |