Normalized defining polynomial
\( x^{25} + 5x - 2 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(99371059376767585012993326250000000000000000000000\) \(\medspace = 2^{22}\cdot 5^{25}\cdot 79496847501414068010394661\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(99.97\) | 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: | not computed | ||
Ramified primes: | \(2\), \(5\), \(79496847501414068010394661\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{39748\!\cdots\!73305}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $a^{7}+a^{5}+a^{3}+a+1$, $18a^{24}-18a^{23}-2a^{22}+23a^{21}-17a^{20}-12a^{19}+32a^{18}-15a^{17}-25a^{16}+38a^{15}-a^{14}-43a^{13}+35a^{12}+21a^{11}-58a^{10}+28a^{9}+39a^{8}-66a^{7}+12a^{6}+69a^{5}-73a^{4}-24a^{3}+110a^{2}-65a+21$, $84a^{24}+28a^{23}+3a^{22}-8a^{21}-13a^{20}-14a^{19}-13a^{18}-9a^{17}-2a^{16}+7a^{15}+16a^{14}+23a^{13}+25a^{12}+22a^{11}+16a^{10}+7a^{9}-2a^{8}-12a^{7}-22a^{6}-32a^{5}-37a^{4}-37a^{3}-26a^{2}-10a+431$, $14a^{24}+12a^{23}+9a^{22}+6a^{21}+3a^{20}-3a^{18}-5a^{17}-8a^{16}-12a^{15}-18a^{14}-22a^{13}-25a^{12}-25a^{11}-25a^{10}-21a^{9}-17a^{8}-13a^{7}-16a^{6}-21a^{5}-29a^{4}-35a^{3}-43a^{2}-46a+27$, $12a^{24}-20a^{23}-a^{22}+30a^{21}+10a^{20}-29a^{19}-19a^{18}+29a^{17}+30a^{16}-28a^{15}-40a^{14}+22a^{13}+50a^{12}-9a^{11}-64a^{10}-a^{9}+69a^{8}+19a^{7}-73a^{6}-42a^{5}+76a^{4}+68a^{3}-76a^{2}-89a+115$, $9a^{24}-5a^{23}+21a^{22}-16a^{21}+5a^{20}-21a^{19}+20a^{18}-a^{17}+16a^{16}-24a^{15}-3a^{14}-3a^{13}+20a^{12}+11a^{11}-16a^{10}-13a^{9}-19a^{8}+47a^{7}-10a^{6}+26a^{5}-74a^{4}+41a^{3}-38a^{2}+105a-37$, $4a^{24}-a^{22}+4a^{21}-6a^{20}-4a^{19}+a^{18}-7a^{17}-3a^{16}-3a^{14}+5a^{13}+2a^{12}-a^{11}+16a^{10}-a^{9}-a^{8}+12a^{7}-2a^{6}-13a^{5}+9a^{4}-19a^{3}-10a+7$, $8a^{24}-5a^{23}+20a^{22}+8a^{21}+7a^{20}+11a^{19}-5a^{18}-2a^{17}+22a^{16}+a^{15}-4a^{14}+11a^{13}-36a^{12}+15a^{11}+7a^{10}-28a^{9}+5a^{8}-29a^{7}-28a^{6}+11a^{5}-16a^{4}-61a^{3}+16a^{2}-58a+25$, $136a^{24}+50a^{23}+9a^{22}+14a^{21}+7a^{20}+31a^{19}+16a^{18}+11a^{17}-11a^{16}-27a^{15}-12a^{14}-25a^{13}+11a^{12}-8a^{11}+32a^{10}+25a^{9}+29a^{8}+12a^{7}-47a^{6}-26a^{5}-65a^{4}+22a^{3}-a^{2}+49a+715$, $68a^{24}+67a^{23}+73a^{22}+49a^{21}+24a^{20}-3a^{19}-52a^{18}-69a^{17}-102a^{16}-110a^{15}-89a^{14}-82a^{13}-19a^{12}+19a^{11}+70a^{10}+134a^{9}+133a^{8}+171a^{7}+135a^{6}+84a^{5}+51a^{4}-72a^{3}-107a^{2}-191a+95$, $20a^{24}-24a^{23}+15a^{22}+21a^{21}-47a^{20}+22a^{19}+25a^{18}-39a^{17}+26a^{16}-12a^{15}-20a^{14}+66a^{13}-56a^{12}-29a^{11}+92a^{10}-63a^{9}-8a^{8}+62a^{7}-92a^{6}+66a^{5}+42a^{4}-139a^{3}+109a^{2}+9a-7$, $19a^{24}+2a^{23}-2a^{22}-27a^{21}-7a^{20}-8a^{19}+11a^{18}+58a^{17}-19a^{16}-12a^{15}-27a^{14}-32a^{13}+36a^{12}-12a^{11}+84a^{10}-2a^{9}-78a^{8}-a^{7}-56a^{6}+64a^{5}+25a^{4}+37a^{3}+62a^{2}-151a+47$ (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: | \( 8545612733538318.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 8545612733538318.0 \cdot 1}{2\cdot\sqrt{99371059376767585012993326250000000000000000000000}}\cr\approx \mathstrut & 3.24542567918385 \end{aligned}\] (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 | R | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $25$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | $24{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
Deg $16$ | $8$ | $2$ | $16$ | ||||
\(5\) | Deg $25$ | $25$ | $1$ | $25$ | |||
\(794\!\cdots\!661\) | $\Q_{79\!\cdots\!61}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |