Normalized defining polynomial
\( x^{26} - x - 4 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(6931174461867292070723506058287666431258711809813849\) \(\medspace = 385720241\cdot 17\!\cdots\!89\) | 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: | \(98.60\) | 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))
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Galois root discriminant: | $385720241^{1/2}17969434126396524964121615951929435902376489^{1/2}\approx 8.32536753655194e+25$ | ||
Ramified primes: | \(385720241\), \(17969\!\cdots\!76489\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{69311\!\cdots\!13849}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
| |
Fundamental units: | $a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+2a+3$, $2a^{25}+4a^{12}-1$, $30a^{25}-38a^{24}+38a^{23}-29a^{22}+13a^{21}+8a^{20}-28a^{19}+44a^{18}-50a^{17}+45a^{16}-29a^{15}+4a^{14}+24a^{13}-49a^{12}+65a^{11}-67a^{10}+54a^{9}-26a^{8}-9a^{7}+47a^{6}-75a^{5}+90a^{4}-84a^{3}+57a^{2}-15a-65$, $a^{23}+2a^{21}-a^{20}+2a^{19}-a^{18}+3a^{17}-a^{16}+3a^{15}-a^{14}+2a^{13}-a^{12}+a^{11}+a^{10}+4a^{8}+6a^{6}-3a^{5}+5a^{4}-3a^{3}+5a^{2}-2a+3$, $4a^{25}+4a^{23}+3a^{22}+2a^{21}+3a^{20}+4a^{19}+a^{18}+6a^{17}+a^{16}+6a^{15}+3a^{14}+6a^{13}+5a^{12}+4a^{11}+5a^{10}+9a^{9}+a^{8}+11a^{7}+5a^{6}+9a^{5}+9a^{4}+8a^{3}+8a^{2}+11a+3$, $a^{25}-2a^{24}-a^{21}+2a^{20}+a^{18}+2a^{17}-a^{16}+a^{15}-a^{14}-3a^{13}+a^{12}-3a^{11}+a^{10}-a^{8}+4a^{7}-a^{6}+a^{5}+3a^{4}-a^{3}+3a^{2}-2a-5$, $9a^{25}-8a^{24}-9a^{23}-15a^{22}+5a^{21}+6a^{20}-a^{19}-19a^{18}-17a^{17}+2a^{16}+9a^{15}+2a^{14}-26a^{13}-21a^{12}-6a^{11}+18a^{10}-2a^{9}-25a^{8}-33a^{7}-11a^{6}+18a^{5}+2a^{4}-21a^{3}-56a^{2}-11a-5$, $5a^{25}-6a^{24}-18a^{23}+2a^{22}+13a^{21}+15a^{20}-10a^{19}-19a^{18}-6a^{17}+19a^{16}+22a^{15}-9a^{14}-23a^{13}-17a^{12}+26a^{11}+26a^{10}+3a^{9}-40a^{8}-19a^{7}+17a^{6}+45a^{5}+6a^{4}-45a^{3}-37a^{2}+14a+57$, $18a^{25}-37a^{24}+51a^{23}-47a^{22}+38a^{21}-20a^{20}-12a^{19}+36a^{18}-54a^{17}+69a^{16}-59a^{15}+37a^{14}-12a^{13}-32a^{12}+66a^{11}-79a^{10}+92a^{9}-73a^{8}+30a^{7}+5a^{6}-60a^{5}+106a^{4}-111a^{3}+116a^{2}-88a-1$, $62a^{25}+45a^{24}-57a^{23}-65a^{22}+50a^{21}+83a^{20}-35a^{19}-102a^{18}+17a^{17}+117a^{16}+11a^{15}-130a^{14}-40a^{13}+133a^{12}+79a^{11}-133a^{10}-116a^{9}+120a^{8}+161a^{7}-97a^{6}-201a^{5}+60a^{4}+240a^{3}-12a^{2}-268a-113$, $10a^{25}+7a^{24}-21a^{23}+16a^{22}-20a^{20}+26a^{19}-7a^{18}-19a^{17}+35a^{16}-15a^{15}-18a^{14}+37a^{13}-25a^{12}-12a^{11}+34a^{10}-36a^{9}+3a^{8}+35a^{7}-49a^{6}+25a^{5}+35a^{4}-60a^{3}+38a^{2}+29a-81$, $6a^{25}+3a^{24}+6a^{23}-a^{22}+2a^{21}-3a^{20}-3a^{19}-6a^{18}-9a^{17}-4a^{16}-3a^{15}+5a^{14}+4a^{13}+8a^{12}+7a^{11}+10a^{10}+7a^{9}+a^{8}-9a^{7}-11a^{6}-16a^{5}-8a^{4}-11a^{3}-5a^{2}+2a+5$, $2a^{25}+12a^{24}+67a^{23}-8a^{22}-7a^{21}-45a^{20}-17a^{19}+27a^{18}-25a^{17}+18a^{16}-29a^{15}+63a^{14}+54a^{13}-6a^{12}-41a^{11}-132a^{10}+17a^{9}+25a^{8}+95a^{7}+27a^{6}-47a^{5}+45a^{4}-28a^{3}+43a^{2}-140a-109$ (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: | \( 12601293459604224 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 12601293459604224 \cdot 1}{2\cdot\sqrt{6931174461867292070723506058287666431258711809813849}}\cr\approx \mathstrut & 1.14604092441957 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 403291461126605635584000000 |
The 2436 conjugacy class representatives for $S_{26}$ |
Character table for $S_{26}$ |
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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/11.8.0.1}{8} }$ | $21{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $17{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | $16{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $26$ |
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 |
---|---|---|---|---|---|---|---|
\(385720241\) | $\Q_{385720241}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(179\!\cdots\!489\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |