Normalized defining polynomial
\( x^{26} - 5x + 3 \)
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: | \(132343681998485749432201649865665728970327492866283857\) \(\medspace = 2483561677\cdot 4048108812907452661\cdot 13163642696128058233262881\) | 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))
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Galois root discriminant: | $2483561677^{1/2}4048108812907452661^{1/2}13163642696128058233262881^{1/2}\approx 3.6379071180898194e+26$ | ||
Ramified primes: | \(2483561677\), \(4048108812907452661\), \(13163642696128058233262881\) | 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: | $\Q(\sqrt{13234\!\cdots\!83857}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $a-1$, $56a^{25}+39a^{24}+17a^{23}+10a^{22}+13a^{21}+a^{20}-a^{19}+9a^{18}-2a^{17}-6a^{16}+9a^{15}-a^{14}-9a^{13}+9a^{12}-12a^{10}+12a^{9}+4a^{8}-17a^{7}+11a^{6}+9a^{5}-21a^{4}+11a^{3}+15a^{2}-27a-271$, $4a^{25}-3a^{24}+a^{23}+3a^{22}-6a^{21}+a^{20}+2a^{19}-7a^{18}+7a^{17}-4a^{16}-6a^{15}+7a^{14}-6a^{13}-2a^{12}+11a^{11}-13a^{10}+7a^{9}+6a^{8}-14a^{7}+15a^{6}+a^{5}-11a^{4}+23a^{3}-14a^{2}-8a+7$, $65a^{25}+38a^{24}+23a^{23}+18a^{22}+11a^{21}+5a^{20}+6a^{19}+4a^{18}+a^{17}+4a^{16}+a^{15}-5a^{14}+a^{12}-6a^{11}-3a^{10}-2a^{9}-8a^{8}+4a^{6}-9a^{5}-4a^{4}+8a^{3}-a^{2}+a-316$, $a^{24}+a^{23}-a^{22}+a^{21}-a^{19}+2a^{18}-a^{17}+a^{15}-2a^{14}+a^{12}-2a^{11}+2a^{10}+a^{9}-2a^{8}+3a^{7}-a^{6}-2a^{5}+5a^{4}-2a^{3}+2a^{2}+7a-5$, $a^{25}-5a^{24}-3a^{23}-2a^{22}+3a^{21}-2a^{20}-a^{19}+3a^{18}+5a^{17}+2a^{16}-4a^{15}+2a^{14}+2a^{12}-13a^{11}+3a^{10}-a^{9}+9a^{8}-14a^{7}+12a^{6}+16a^{4}-19a^{3}+8a^{2}-8a+10$, $10a^{25}+6a^{24}-2a^{23}-8a^{22}-12a^{21}-18a^{20}-25a^{19}-26a^{18}-26a^{17}-28a^{16}-27a^{15}-18a^{14}-13a^{13}-9a^{12}+2a^{11}+15a^{10}+19a^{9}+25a^{8}+38a^{7}+42a^{6}+36a^{5}+40a^{4}+40a^{3}+29a^{2}+12a-38$, $5a^{25}+8a^{24}+8a^{23}+4a^{22}-6a^{21}-11a^{20}-15a^{19}-3a^{18}+5a^{17}+15a^{16}+13a^{15}+6a^{14}-3a^{13}-18a^{12}-15a^{11}-11a^{10}+13a^{9}+19a^{8}+23a^{7}+12a^{6}-9a^{5}-20a^{4}-36a^{3}-8a^{2}+4a+10$, $4a^{25}-a^{24}-2a^{23}-4a^{22}-2a^{21}+5a^{20}+a^{19}+4a^{18}+a^{17}-9a^{16}-4a^{15}+10a^{13}+9a^{12}-3a^{11}-11a^{10}-17a^{9}+11a^{8}+19a^{7}+4a^{6}-a^{5}-26a^{4}-9a^{3}+17a^{2}+16a-11$, $a^{25}-8a^{24}+5a^{23}-6a^{22}+10a^{21}+a^{20}+17a^{19}+6a^{18}+9a^{17}-9a^{16}-9a^{15}-15a^{14}-11a^{13}-6a^{11}+8a^{10}-6a^{9}+25a^{8}-a^{7}+42a^{6}-15a^{5}+20a^{4}-54a^{3}+13a^{2}-60a+35$, $10a^{25}-11a^{24}-7a^{23}-12a^{22}+16a^{21}+25a^{20}-11a^{19}-36a^{18}+9a^{17}+29a^{16}+3a^{15}+2a^{14}-37a^{13}-14a^{12}+50a^{11}+36a^{10}-54a^{9}-34a^{8}+28a^{7}+15a^{6}+51a^{5}-30a^{4}-88a^{3}+16a^{2}+118a-58$, $4a^{25}-16a^{24}+55a^{23}-59a^{22}+35a^{21}-18a^{20}-45a^{19}+74a^{18}-84a^{17}+78a^{16}-49a^{14}+109a^{13}-149a^{12}+77a^{11}-32a^{10}-70a^{9}+188a^{8}-160a^{7}+159a^{6}-60a^{5}-136a^{4}+191a^{3}-285a^{2}+264a-71$, $27a^{25}-6a^{24}-10a^{23}+39a^{22}-2a^{21}-25a^{20}+35a^{19}-9a^{18}-51a^{17}+31a^{16}-a^{15}-62a^{14}+46a^{13}+34a^{12}-62a^{11}+57a^{10}+62a^{9}-88a^{8}+28a^{7}+66a^{6}-129a^{5}-7a^{4}+100a^{3}-129a^{2}-9a+38$ (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: | \( 22704789684328616 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 22704789684328616 \cdot 2}{2\cdot\sqrt{132343681998485749432201649865665728970327492866283857}}\cr\approx \mathstrut & 0.945114200125254 \end{aligned}\] (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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $22{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $19{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2483561677\) | $\Q_{2483561677}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(4048108812907452661\) | $\Q_{4048108812907452661}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(131\!\cdots\!881\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |