Normalized defining polynomial
\( x^{25} + 4x - 3 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1526737768847242991861919888652736835584204673849\) \(\medspace = 3^{24}\cdot 54\!\cdots\!29\) | 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: | \(84.60\) | 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: | not computed | ||
Ramified primes: | \(3\), \(54057\!\cdots\!43929\) | 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: | $\Q(\sqrt{54057\!\cdots\!43929}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$
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: | $a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+2a^{8}-2a^{7}+2a^{6}-2a^{5}+2a^{4}-2a^{3}+2a^{2}-2a+1$, $2a^{18}-a^{17}-3a^{10}+2a^{9}+5a^{2}-5a+1$, $2a^{24}+3a^{23}+4a^{22}+2a^{21}+2a^{20}+6a^{19}-2a^{18}+6a^{17}+a^{16}-a^{15}+6a^{14}-3a^{13}+a^{12}+a^{11}-5a^{9}+5a^{8}-8a^{7}-a^{6}+5a^{5}-17a^{4}+10a^{3}-9a^{2}-5a+10$, $2a^{24}-a^{23}+a^{22}+a^{20}-2a^{18}+2a^{17}-3a^{16}+a^{15}+5a^{14}-a^{13}-2a^{12}-6a^{11}+3a^{10}+6a^{9}-3a^{8}+4a^{7}-5a^{6}-5a^{5}+3a^{4}+2a^{3}+10a^{2}-11a+4$, $a^{24}-a^{21}+2a^{20}-a^{19}-a^{17}+2a^{16}-a^{15}-a^{13}+2a^{12}-a^{11}-a^{10}+a^{9}+a^{8}-2a^{7}-2a^{6}+5a^{5}-2a^{4}-2a^{3}-a^{2}+7a-4$, $19a^{24}+13a^{23}+8a^{22}+11a^{21}+3a^{20}+4a^{19}+6a^{18}-a^{17}+3a^{16}+4a^{15}-4a^{14}+4a^{13}+2a^{12}-6a^{11}+8a^{10}-4a^{9}-2a^{8}+5a^{7}-2a^{6}-6a^{5}+14a^{4}-16a^{3}+8a^{2}+3a+67$, $10a^{24}+11a^{23}+12a^{22}+11a^{21}+11a^{20}+10a^{19}+11a^{18}+13a^{17}+14a^{16}+16a^{15}+13a^{14}+13a^{13}+8a^{12}+7a^{11}+3a^{10}+a^{9}+2a^{8}+5a^{6}-2a^{5}-11a^{3}-10a^{2}-16a+25$, $7a^{24}+10a^{23}+11a^{22}+6a^{21}+4a^{20}+5a^{19}+2a^{18}-6a^{17}-5a^{16}-2a^{15}-7a^{14}-9a^{13}-2a^{12}+5a^{11}-2a^{10}+3a^{9}+12a^{8}+11a^{7}+4a^{5}+11a^{4}-5a^{3}-12a^{2}-4a+28$, $8a^{24}-3a^{23}-9a^{22}+4a^{21}+12a^{20}-10a^{18}-3a^{17}+10a^{16}+6a^{15}-13a^{14}-11a^{13}+9a^{12}+11a^{11}-7a^{10}-16a^{9}+4a^{8}+25a^{7}+a^{6}-28a^{5}+a^{4}+32a^{3}+7a^{2}-34a+11$, $3a^{24}+2a^{23}+3a^{22}+4a^{21}+2a^{20}+6a^{19}-2a^{18}+2a^{17}-5a^{16}-a^{15}-5a^{14}-7a^{13}-a^{12}-6a^{11}+6a^{10}-6a^{9}+8a^{8}+2a^{7}+8a^{6}+6a^{5}-4a^{4}+11a^{3}-10a^{2}+8a-7$, $5a^{24}+12a^{23}+6a^{22}-5a^{21}-10a^{20}-11a^{19}+14a^{17}+12a^{16}+3a^{15}-9a^{14}-18a^{13}-7a^{12}+7a^{11}+15a^{10}+19a^{9}-2a^{8}-22a^{7}-17a^{6}-8a^{5}+14a^{4}+33a^{3}+12a^{2}-12a-10$, $14a^{24}+17a^{23}+16a^{22}+8a^{21}-6a^{20}-13a^{19}-3a^{18}+12a^{17}+15a^{16}+9a^{15}-a^{14}-14a^{13}-20a^{12}-3a^{11}+22a^{10}+23a^{9}+3a^{8}-14a^{7}-20a^{6}-19a^{5}+3a^{4}+33a^{3}+30a^{2}-6a+25$ (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: | \( 310216853513864.2 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 310216853513864.2 \cdot 1}{2\cdot\sqrt{1526737768847242991861919888652736835584204673849}}\cr\approx \mathstrut & 0.950476938771731 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ |
Character table for $S_{25}$ |
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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $24{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.12.12.12 | $x^{12} - 12 x^{11} + 156 x^{10} - 1104 x^{9} + 3420 x^{8} + 972 x^{7} - 1890 x^{6} + 1404 x^{5} + 1944 x^{4} + 432 x^{3} + 324 x^{2} + 324 x + 81$ | $3$ | $4$ | $12$ | 12T41 | $[3/2, 3/2]_{2}^{4}$ | |
3.12.12.14 | $x^{12} + 54 x^{10} - 240 x^{9} - 1935 x^{8} + 7236 x^{7} + 45198 x^{6} + 60156 x^{5} + 16686 x^{4} - 13824 x^{3} + 5184 x^{2} - 972 x + 81$ | $3$ | $4$ | $12$ | 12T173 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
\(540\!\cdots\!929\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |