Normalized defining polynomial
\( x^{25} - 3 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(25084781938833344838712946511805057525634765625\) \(\medspace = 3^{24}\cdot 5^{50}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(71.78\) | 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: | $3^{24/25}5^{1083/500}\approx 93.75737378401558$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $a^{15}+a^{5}+1$, $a^{13}-a-1$, $a^{20}-a^{15}+a^{10}-2$, $a^{24}-a^{22}+a^{21}-2a^{20}+2a^{19}-2a^{18}+2a^{17}-2a^{16}+a^{15}-a^{13}+3a^{12}-4a^{11}+4a^{10}-4a^{9}+2a^{8}-a^{7}-a^{6}+2a^{5}-3a^{4}+4a^{3}-5a^{2}+5a-4$, $a^{24}-a^{21}+a^{18}-a^{15}+2a^{11}-a^{10}-2a^{8}+a^{7}+2a^{5}-a^{4}-2a^{2}+a-1$, $2a^{24}+a^{23}-a^{22}-3a^{21}-a^{20}+a^{19}-a^{18}-3a^{17}-3a^{16}+3a^{14}+3a^{13}-2a^{11}+2a^{10}+4a^{9}+a^{8}-2a^{7}-4a^{6}-a^{5}+4a^{4}+3a^{3}-2a^{2}-2a+4$, $2a^{24}+2a^{23}-a^{22}-2a^{21}+2a^{19}-3a^{17}+4a^{15}+2a^{14}-2a^{13}-3a^{12}+2a^{10}-a^{9}-a^{8}+3a^{7}+2a^{6}-a^{5}-a^{4}-3a-2$, $6a^{24}+9a^{23}-10a^{22}-6a^{21}+14a^{20}+a^{19}-15a^{18}+5a^{17}+15a^{16}-11a^{15}-12a^{14}+17a^{13}+6a^{12}-20a^{11}+a^{10}+22a^{9}-10a^{8}-20a^{7}+19a^{6}+14a^{5}-26a^{4}-7a^{3}+31a^{2}-6a-32$, $8a^{24}+3a^{23}-6a^{22}-7a^{21}+3a^{20}+9a^{19}+3a^{18}-9a^{17}-9a^{16}+3a^{15}+12a^{14}+4a^{13}-10a^{12}-11a^{11}+5a^{10}+15a^{9}+6a^{8}-13a^{7}-13a^{6}+4a^{5}+18a^{4}+4a^{3}-17a^{2}-18a+7$, $3a^{24}+a^{23}-4a^{22}+3a^{21}+a^{20}-7a^{19}+9a^{18}-8a^{17}+3a^{16}+a^{15}-a^{14}-3a^{13}+10a^{12}-14a^{11}+13a^{10}-7a^{9}-a^{8}+3a^{7}-10a^{5}+16a^{4}-15a^{3}+5a^{2}+8a-13$, $2a^{24}-3a^{23}+a^{22}-2a^{21}+2a^{20}+2a^{18}-2a^{17}-2a^{16}-a^{15}+a^{14}+3a^{13}-6a^{10}+2a^{9}-a^{8}+6a^{7}-3a^{6}+3a^{5}-9a^{4}+5a^{3}-4a^{2}+7a-4$, $2a^{24}+2a^{23}+3a^{22}-a^{20}-2a^{19}-5a^{18}-9a^{17}-8a^{16}-4a^{15}-9a^{14}-12a^{13}-7a^{12}-4a^{11}-4a^{10}-3a^{9}+3a^{8}+4a^{7}+4a^{6}+13a^{5}+14a^{4}+9a^{3}+11a^{2}+19a+19$ (assuming GRH) | 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: | \( 40725438575985.81 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 40725438575985.81 \cdot 1}{2\cdot\sqrt{25084781938833344838712946511805057525634765625}}\cr\approx \mathstrut & 0.973462049600231 \end{aligned}\] (assuming GRH)
Galois group
$C_{25}:C_{20}$ (as 25T40):
A solvable group of order 500 |
The 26 conjugacy class representatives for $C_{25}:C_{20}$ |
Character table for $C_{25}:C_{20}$ is not computed |
Intermediate fields
5.1.253125.1 |
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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $25$ | $20{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $25$ | $20{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $25$ | $25$ | $1$ | $24$ | |||
\(5\) | Deg $25$ | $25$ | $1$ | $50$ |