Normalized defining polynomial
\( x^{26} - 2x + 3 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-5216004038214069684678365528327600202444877856768\) \(\medspace = -\,2^{27}\cdot 47\cdot 349\cdot 1395923\cdot 415496197\cdot 4084851144240798246517\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(74.77\) | 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: | \(2\), \(47\), \(349\), \(1395923\), \(415496197\), \(4084851144240798246517\) | 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: | $\Q(\sqrt{-77724\!\cdots\!07362}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $12$ | 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: | $a^{25}-a^{23}+a^{21}+a^{20}-a^{16}+a^{15}+2a^{14}-a^{12}-a^{11}+a^{10}+a^{8}+2a^{7}-2a^{6}-2a^{5}+2a^{3}+a^{2}-a-1$, $a^{24}+a^{23}-a^{22}-2a^{21}+2a^{19}+a^{18}-a^{17}-a^{16}+a^{12}+a^{11}-a^{10}-2a^{9}+2a^{7}+a^{6}-a^{5}-a^{4}+1$, $a^{24}+2a^{23}+2a^{22}+a^{21}+a^{18}+2a^{17}+2a^{16}+a^{15}+a^{12}+2a^{11}+2a^{10}+a^{9}+a^{6}+2a^{5}+2a^{4}+a^{3}+1$, $9a^{25}+10a^{24}+10a^{23}+11a^{22}+11a^{21}+11a^{20}+12a^{19}+11a^{18}+12a^{17}+11a^{16}+10a^{15}+10a^{14}+8a^{13}+9a^{12}+7a^{11}+6a^{10}+6a^{9}+3a^{8}+3a^{7}+a^{6}-a^{5}-a^{4}-3a^{3}-3a^{2}-5a-25$, $3a^{25}+3a^{24}+2a^{23}+a^{22}-2a^{21}-5a^{20}-3a^{19}+2a^{18}+3a^{17}+3a^{16}+4a^{15}-6a^{13}-6a^{12}-a^{11}+a^{10}+3a^{9}+7a^{8}+4a^{7}-4a^{6}-6a^{5}-3a^{4}-2a^{3}+9a+2$, $2a^{25}+a^{24}+2a^{23}-2a^{22}-2a^{20}-2a^{19}+a^{18}-2a^{17}+3a^{16}+a^{15}+a^{14}+4a^{13}-2a^{12}+2a^{11}-2a^{10}-4a^{9}+a^{8}-6a^{7}+3a^{6}+a^{4}+7a^{3}-3a^{2}+5a-8$, $11a^{25}+2a^{24}-11a^{23}-4a^{22}+12a^{21}+5a^{20}-13a^{19}-6a^{18}+14a^{17}+9a^{16}-17a^{15}-8a^{14}+17a^{13}+11a^{12}-19a^{11}-12a^{10}+19a^{9}+14a^{8}-19a^{7}-19a^{6}+21a^{5}+22a^{4}-24a^{3}-24a^{2}+24a+8$, $8a^{25}+7a^{24}+8a^{23}+10a^{22}+9a^{21}+9a^{20}+10a^{19}+10a^{18}+10a^{17}+8a^{16}+10a^{15}+10a^{14}+5a^{13}+9a^{12}+8a^{11}+3a^{10}+6a^{9}+4a^{8}+3a^{7}+a^{6}-a^{5}+3a^{4}-3a^{3}-6a^{2}-20$, $3a^{25}+3a^{24}+5a^{23}+5a^{22}+2a^{21}+a^{20}+2a^{19}+a^{18}-2a^{17}-3a^{16}+2a^{14}-a^{12}+a^{11}+a^{10}-3a^{9}-6a^{8}-4a^{7}-2a^{6}-6a^{5}-8a^{4}-2a^{3}+2a^{2}-a-11$, $a^{25}-a^{22}-3a^{21}-2a^{20}+a^{19}-2a^{17}+4a^{15}+2a^{14}-3a^{13}-5a^{12}+a^{10}-4a^{9}-7a^{8}+a^{7}+5a^{6}+a^{5}-3a^{4}+a^{3}+2a^{2}-a-7$, $a^{25}-12a^{23}+12a^{22}-12a^{21}+7a^{20}+4a^{19}-10a^{18}+21a^{17}-14a^{16}+12a^{15}+4a^{14}-14a^{13}+25a^{12}-23a^{11}+14a^{10}-a^{9}-20a^{8}+30a^{7}-36a^{6}+20a^{5}-5a^{4}-25a^{3}+43a^{2}-46a+32$, $a^{25}+7a^{24}+a^{23}+4a^{22}-2a^{21}-3a^{20}-4a^{19}-5a^{18}+a^{17}-6a^{16}+10a^{15}+2a^{14}+6a^{13}+4a^{12}+5a^{11}-2a^{10}-10a^{9}+6a^{8}-15a^{7}+2a^{6}-2a^{5}+9a^{4}+a^{3}+7a^{2}+13a-13$ (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: | \( 228191453653879.84 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 228191453653879.84 \cdot 1}{2\cdot\sqrt{5216004038214069684678365528327600202444877856768}}\cr\approx \mathstrut & 1.18833450720332 \end{aligned}\] (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 | R | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | $19{,}\,{\href{/padicField/37.7.0.1}{7} }$ | $22{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $21{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | R | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
Deg $24$ | $2$ | $12$ | $24$ | ||||
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.5.0.1 | $x^{5} + x + 42$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
47.18.0.1 | $x^{18} + 6 x^{11} + 41 x^{10} + 42 x^{9} + 26 x^{8} + 44 x^{7} + 24 x^{6} + 22 x^{5} + 11 x^{4} + 5 x^{3} + 45 x^{2} + 33 x + 5$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(349\) | $\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(1395923\) | $\Q_{1395923}$ | $x$ | $1$ | $1$ | $0$ | 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 $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(415496197\) | $\Q_{415496197}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{415496197}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(408\!\cdots\!517\) | $\Q_{40\!\cdots\!17}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |