Normalized defining polynomial
\( x^{26} - 3x - 3 \)
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: | \(5441773036128047327289282046933845720175590747393\) \(\medspace = 3^{25}\cdot 19\cdot 9619\cdot 306629767564201\cdot 114607019135083291\) | 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: | \(74.89\) | 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: | $3^{25/26}19^{1/2}9619^{1/2}306629767564201^{1/2}114607019135083291^{1/2}\approx 7.288278598531036e+18$ | ||
Ramified primes: | \(3\), \(19\), \(9619\), \(306629767564201\), \(114607019135083291\) | 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: | $\Q(\sqrt{19267\!\cdots\!01953}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a+1$, $a^{25}-a^{24}-a^{22}+a^{21}-a^{18}+a^{15}-a^{12}+a^{11}+a^{10}+a^{9}-a^{8}+a^{6}+a^{5}-a^{3}-a^{2}-2$, $a^{24}+a^{23}-a^{21}-a^{20}-a^{19}+a^{18}+a^{17}+a^{16}-a^{15}-a^{14}-a^{13}+a^{12}+2a^{11}+2a^{10}-a^{9}-2a^{8}-3a^{7}-a^{6}+2a^{5}+3a^{4}+a^{3}-2a^{2}-3a-1$, $a^{24}+a^{21}+a^{20}-a^{19}+2a^{18}+a^{15}+a^{14}+a^{13}+2a^{11}+3a^{8}+a^{7}+2a^{5}+2a^{4}-a^{3}+4a^{2}+3a-1$, $a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}-a^{16}-a^{15}-a^{14}-2a^{13}-2a^{12}-3a^{11}-3a^{10}-3a^{9}-3a^{8}-2a^{7}-2a^{6}+a^{3}+2a^{2}+3a+2$, $a^{25}-a^{23}+2a^{21}-a^{19}+2a^{17}+2a^{16}-2a^{15}+a^{13}+3a^{12}-2a^{11}-a^{10}+2a^{9}+4a^{8}-3a^{6}+a^{5}+4a^{4}+a^{3}-3a^{2}+4$, $a^{16}+a^{13}-a^{11}+a^{10}-a^{8}+a^{6}-a^{5}-2a^{4}+a^{3}-a^{2}-2a+1$, $2a^{25}-4a^{24}+4a^{23}-2a^{22}-2a^{21}+6a^{20}-6a^{19}+3a^{18}+a^{17}-5a^{16}+8a^{15}-7a^{14}+4a^{13}-6a^{11}+11a^{10}-10a^{9}+4a^{8}+2a^{7}-9a^{6}+11a^{5}-7a^{4}+a^{3}+2a^{2}-7a+4$, $2a^{25}-3a^{24}+2a^{22}-a^{21}-a^{19}+3a^{17}-3a^{16}+a^{15}+a^{14}+a^{12}-3a^{11}+3a^{10}+3a^{9}-4a^{8}-a^{6}+5a^{5}-8a^{3}+a^{2}+3a-4$, $6a^{25}-4a^{24}+2a^{23}-a^{21}+2a^{20}-3a^{19}+5a^{18}-7a^{17}+9a^{16}-9a^{15}+9a^{14}-8a^{13}+6a^{12}-3a^{11}+a^{10}-a^{9}+2a^{8}-3a^{7}+3a^{6}-4a^{5}+3a^{4}-a^{3}-3a^{2}+6a-25$, $5a^{25}+4a^{24}+3a^{23}-a^{22}-a^{21}-4a^{20}-3a^{19}+a^{18}+6a^{17}+6a^{16}+5a^{15}+3a^{14}-2a^{13}-7a^{12}-6a^{11}+2a^{10}+6a^{9}+11a^{8}+13a^{7}+10a^{6}-3a^{5}-11a^{4}-11a^{3}-6a^{2}+a+2$, $4a^{25}-4a^{24}+3a^{23}+a^{21}-3a^{20}+2a^{19}-2a^{18}+a^{17}+2a^{16}-3a^{14}-a^{13}+3a^{12}-4a^{11}+6a^{10}+2a^{9}-4a^{8}-5a^{7}+3a^{6}-3a^{5}+3a^{4}+6a^{3}+2a^{2}-9a-13$, $5a^{25}+2a^{24}-2a^{23}-a^{22}-5a^{21}+2a^{20}+4a^{19}+2a^{18}+a^{17}-7a^{16}-4a^{15}+7a^{13}+9a^{12}-6a^{10}-13a^{9}-6a^{8}+8a^{7}+12a^{6}+13a^{5}-2a^{4}-18a^{3}-12a^{2}-4a-2$ (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: | \( 326818418751639.6 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 326818418751639.6 \cdot 1}{2\cdot\sqrt{5441773036128047327289282046933845720175590747393}}\cr\approx \mathstrut & 1.06077854729640 \end{aligned}\] (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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | $19{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | $25{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $26$ | $16{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $26$ | $26$ | $1$ | $25$ | |||
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(9619\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(306629767564201\) | $\Q_{306629767564201}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(114607019135083291\) | $\Q_{114607019135083291}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |