Normalized defining polynomial
\( x^{25} + 3x - 1 \)
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: | \(1130059131820674428173933522818307867264642393\) \(\medspace = 7\cdot 14691751\cdot 10\!\cdots\!49\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(63.41\) | 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: | $7^{1/2}14691751^{1/2}10988276266833335135516653653549317449^{1/2}\approx 3.3616352149224554e+22$ | ||
Ramified primes: | \(7\), \(14691751\), \(10988\!\cdots\!17449\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{11300\!\cdots\!42393}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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Fundamental units: | $a$, $a^{13}-2a^{7}+2a$, $a^{24}-a^{20}+a^{18}-a^{14}-a^{13}+a^{12}+a^{11}+a^{9}-2a^{7}+a^{5}-a^{4}+a^{3}+a^{2}-2a+2$, $a^{20}+a^{19}+a^{18}+a^{17}+a^{15}-a^{13}-a^{12}-a^{11}-a^{10}-a^{9}-a^{8}-a^{7}+2a^{6}+a^{4}+2a^{3}+a^{2}+3a$, $a^{23}+2a^{22}+2a^{21}-2a^{19}-a^{18}+a^{17}+a^{16}-a^{15}-3a^{14}-2a^{13}+a^{12}+2a^{11}+a^{10}-a^{9}-2a^{8}+a^{7}+4a^{6}+2a^{5}-2a^{4}-4a^{3}-2a^{2}+2a+1$, $a^{24}+a^{22}+a^{21}-a^{20}-a^{19}-a^{18}-a^{17}+a^{14}+2a^{13}-2a^{9}-a^{8}+2a^{5}+a^{4}+a^{3}+2a^{2}-3a$, $a^{23}-a^{21}+a^{18}-a^{15}+a^{13}+a^{11}-2a^{10}+2a^{7}-2a^{5}-a^{3}+3a^{2}-1$, $2a^{24}-4a^{22}-a^{21}+4a^{20}-4a^{18}-a^{17}+2a^{16}+2a^{15}-3a^{14}-5a^{13}+4a^{12}+5a^{11}-5a^{10}-5a^{9}+a^{8}+6a^{7}+3a^{6}-10a^{5}-4a^{4}+11a^{3}+4a^{2}-6a$, $a^{22}+3a^{21}+4a^{20}+3a^{19}+2a^{18}+2a^{17}+2a^{16}-3a^{14}-4a^{13}-3a^{12}-3a^{11}-4a^{10}-3a^{9}+2a^{7}+2a^{6}+2a^{5}+4a^{4}+5a^{3}+3a^{2}-a-1$, $3a^{24}+a^{23}+4a^{21}-4a^{20}+2a^{19}-3a^{18}-3a^{17}+3a^{16}-4a^{15}+5a^{14}+2a^{13}-2a^{12}+7a^{11}-8a^{10}+2a^{9}-2a^{8}-7a^{7}+8a^{6}-6a^{5}+7a^{4}+4a^{3}-5a^{2}+9a-2$, $3a^{24}-2a^{23}+6a^{22}-2a^{21}+3a^{20}-7a^{19}+a^{18}-4a^{17}+8a^{16}-2a^{15}+6a^{14}-6a^{13}-a^{12}-5a^{11}+2a^{10}+5a^{9}+3a^{8}+a^{7}-5a^{6}-2a^{5}-7a^{4}+8a^{3}-2a^{2}+16a-6$, $2a^{24}+a^{23}+2a^{21}+2a^{18}-2a^{17}-a^{16}+2a^{15}-3a^{14}+a^{12}-4a^{11}+2a^{10}-4a^{8}+3a^{7}-2a^{6}-2a^{5}+2a^{4}-6a^{3}-a^{2}+2a-1$ (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)]
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Regulator: | \( 4423868533942.698 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 4423868533942.698 \cdot 1}{2\cdot\sqrt{1130059131820674428173933522818307867264642393}}\cr\approx \mathstrut & 0.498207255662346 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ are not computed |
Character table for $S_{25}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $19{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $21{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.23.0.1 | $x^{23} + 4 x^{2} + 4 x + 4$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(14691751\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(109\!\cdots\!449\) | $\Q_{10\!\cdots\!49}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |