Normalized defining polynomial
\( x^{26} - 5x + 2 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(132348897801919332141495784832764183334256911356205393\) \(\medspace = 257\cdot 68112155980159\cdot 75\!\cdots\!11\) | 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: | \(110.44\) | 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: | $257^{1/2}68112155980159^{1/2}7560709958859746994536436073044299311^{1/2}\approx 3.637978804252704e+26$ | ||
Ramified primes: | \(257\), \(68112155980159\), \(75607\!\cdots\!99311\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{13234\!\cdots\!05393}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | 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: | $a^{25}-7a^{24}-3a^{23}+8a^{22}+7a^{21}-10a^{20}-4a^{19}+11a^{18}+8a^{17}-7a^{16}-12a^{15}+13a^{14}+13a^{13}-13a^{12}-13a^{11}+6a^{10}+20a^{9}-13a^{8}-28a^{7}+12a^{6}+19a^{5}-7a^{4}-35a^{3}+a^{2}+43a-17$, $a^{25}-a^{24}+a^{23}-a^{22}+a^{21}-a^{20}+a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+a^{3}-3a+1$, $5a^{25}+5a^{24}-9a^{23}+4a^{22}+6a^{21}-11a^{20}+a^{19}+10a^{18}-12a^{17}-2a^{16}+16a^{15}-11a^{14}-6a^{13}+22a^{12}-13a^{11}-10a^{10}+24a^{9}-13a^{8}-18a^{7}+26a^{6}-4a^{5}-31a^{4}+34a^{3}+6a^{2}-41a+13$, $17a^{25}-15a^{24}+11a^{23}-6a^{22}+8a^{20}-18a^{19}+26a^{18}-29a^{17}+28a^{16}-26a^{15}+20a^{14}-6a^{13}-13a^{12}+29a^{11}-41a^{10}+50a^{9}-56a^{8}+56a^{7}-44a^{6}+18a^{5}+10a^{4}-35a^{3}+64a^{2}-96a+29$, $a^{25}-2a^{23}-6a^{22}-7a^{21}-a^{20}+6a^{19}+7a^{18}+4a^{17}+a^{16}-2a^{15}-7a^{14}-10a^{13}-4a^{12}+7a^{11}+12a^{10}+7a^{9}-3a^{7}-8a^{6}-15a^{5}-11a^{4}+9a^{3}+23a^{2}+13a-5$, $3a^{25}+11a^{24}+12a^{23}+5a^{22}-8a^{21}-21a^{20}-24a^{19}-14a^{18}+4a^{17}+24a^{16}+36a^{15}+30a^{14}+9a^{13}-17a^{12}-38a^{11}-40a^{10}-20a^{9}+7a^{8}+28a^{7}+35a^{6}+20a^{5}-7a^{4}-25a^{3}-27a^{2}-9a+11$, $24a^{25}+21a^{24}+17a^{23}+12a^{22}+13a^{21}+14a^{20}+18a^{19}+19a^{18}+15a^{17}+14a^{16}+17a^{15}+26a^{14}+21a^{13}+18a^{12}+20a^{11}+20a^{10}+31a^{9}+31a^{8}+19a^{7}+23a^{6}+29a^{5}+40a^{4}+33a^{3}+29a^{2}+28a-89$, $4a^{25}+a^{24}-4a^{23}-a^{22}+5a^{21}+2a^{20}-5a^{19}-2a^{18}+7a^{17}+4a^{16}-7a^{15}-5a^{14}+7a^{13}+7a^{12}-7a^{11}-9a^{10}+6a^{9}+9a^{8}-8a^{7}-12a^{6}+7a^{5}+13a^{4}-7a^{3}-18a^{2}+5a+1$, $26a^{25}-52a^{24}-59a^{23}+17a^{22}+76a^{21}+35a^{20}-64a^{19}-85a^{18}+22a^{17}+117a^{16}+44a^{15}-109a^{14}-109a^{13}+56a^{12}+151a^{11}+36a^{10}-140a^{9}-135a^{8}+68a^{7}+211a^{6}+58a^{5}-220a^{4}-195a^{3}+145a^{2}+300a-121$, $47a^{25}-96a^{24}+30a^{23}+77a^{22}-113a^{21}+11a^{20}+114a^{19}-126a^{18}-20a^{17}+157a^{16}-131a^{15}-66a^{14}+205a^{13}-124a^{12}-130a^{11}+256a^{10}-99a^{9}-213a^{8}+304a^{7}-50a^{6}-315a^{5}+341a^{4}+31a^{3}-436a^{2}+358a-81$, $2a^{25}-a^{24}+2a^{23}-a^{22}-a^{21}+a^{20}-4a^{19}+3a^{18}+2a^{17}-5a^{16}+4a^{15}-a^{14}+a^{13}+4a^{12}-8a^{11}+2a^{10}+3a^{9}-5a^{8}+a^{7}-2a^{6}-a^{5}+5a^{4}-a^{3}-3a^{2}+7a-11$, $7a^{24}+2a^{23}-2a^{22}-10a^{21}+3a^{20}+7a^{19}+6a^{18}-9a^{17}-7a^{16}-2a^{15}+18a^{14}+a^{13}-10a^{12}-13a^{11}+6a^{10}+15a^{9}+9a^{8}-20a^{7}-14a^{6}+8a^{5}+19a^{4}+13a^{3}-27a^{2}-20a+11$, $15a^{25}+5a^{24}+7a^{23}-7a^{22}-10a^{21}-a^{20}+18a^{18}+5a^{17}-10a^{16}-10a^{15}-15a^{14}+16a^{13}+14a^{12}+4a^{11}+4a^{10}-31a^{9}-3a^{8}-a^{7}+18a^{6}+38a^{5}-28a^{4}-11a^{3}-35a^{2}+6a-23$ (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: | \( 55213926480331360 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 55213926480331360 \cdot 1}{2\cdot\sqrt{132348897801919332141495784832764183334256911356205393}}\cr\approx \mathstrut & 1.14915043069212 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 403291461126605635584000000 |
The 2436 conjugacy class representatives for $S_{26}$ are not computed |
Character table for $S_{26}$ 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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $24{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $26$ | $22{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $15{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(257\) | $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(68112155980159\) | $\Q_{68112155980159}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(756\!\cdots\!311\) | $\Q_{75\!\cdots\!11}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |