Normalized defining polynomial
\( x^{25} + 4x - 1 \)
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: | \(1501652986908498464865176954664165668592017173849\) \(\medspace = 38117394317861\cdot 39\!\cdots\!09\) | 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: | \(84.54\) | 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: | $38117394317861^{1/2}39395478462830178925440533821507109^{1/2}\approx 1.225419514659571e+24$ | ||
Ramified primes: | \(38117394317861\), \(39395\!\cdots\!07109\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{15016\!\cdots\!73849}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
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
$C_{2}$, which has order $2$ (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: | $a$, $a^{13}+2a^{7}+2a$, $a^{24}+3a^{23}+a^{22}-a^{21}-2a^{20}-3a^{19}+a^{18}+3a^{17}+3a^{16}+2a^{15}-3a^{14}-4a^{13}-3a^{12}+5a^{10}+4a^{9}+a^{8}-4a^{7}-6a^{6}-2a^{5}+2a^{4}+9a^{3}+6a^{2}-a-1$, $7a^{24}+3a^{23}-5a^{22}-7a^{21}-a^{20}+7a^{19}+8a^{18}-2a^{17}-12a^{16}-6a^{15}+10a^{14}+13a^{13}-2a^{12}-14a^{11}-8a^{10}+7a^{9}+14a^{8}+6a^{7}-11a^{6}-18a^{5}-a^{4}+22a^{3}+17a^{2}-13a+1$, $3a^{24}+6a^{23}+2a^{22}+3a^{21}+7a^{20}+4a^{19}+2a^{18}+8a^{17}+5a^{16}+a^{15}+7a^{14}+6a^{13}-a^{12}+4a^{11}+7a^{10}-5a^{9}+a^{8}+5a^{7}-7a^{6}-5a^{5}+2a^{4}-7a^{3}-15a^{2}+2a+2$, $3a^{23}-3a^{22}+3a^{21}+2a^{20}-4a^{19}+5a^{18}-3a^{16}+4a^{15}-3a^{14}-4a^{13}+6a^{12}-6a^{11}-2a^{10}+7a^{9}-10a^{8}+5a^{7}+5a^{6}-8a^{5}+9a^{4}+2a^{3}-9a^{2}+13a-1$, $3a^{24}+6a^{23}+4a^{22}+5a^{21}+a^{20}+a^{19}-3a^{18}-4a^{17}-8a^{16}-8a^{15}-8a^{14}-6a^{13}-3a^{12}+6a^{10}+7a^{9}+12a^{8}+9a^{7}+14a^{6}+7a^{5}+10a^{4}-3a^{3}-a^{2}-15a+3$, $6a^{24}-6a^{23}+5a^{22}-3a^{21}+6a^{20}-6a^{19}+5a^{18}-a^{17}+3a^{16}-4a^{15}+3a^{14}+3a^{13}-4a^{12}+2a^{11}-2a^{10}+8a^{9}-13a^{8}+11a^{7}-10a^{6}+14a^{5}-23a^{4}+21a^{3}-18a^{2}+17a-5$, $a^{24}+a^{23}+a^{22}-6a^{21}+4a^{20}+a^{19}-3a^{18}+3a^{17}+2a^{16}+4a^{15}-4a^{14}-2a^{13}+6a^{12}-7a^{11}-4a^{10}+6a^{9}+3a^{7}-6a^{6}+8a^{5}+4a^{4}-20a^{3}+9a^{2}-4a+2$, $a^{24}+2a^{23}+a^{22}+2a^{21}+6a^{20}+7a^{19}+6a^{18}+6a^{17}+9a^{16}+13a^{15}+11a^{14}+10a^{13}+13a^{12}+17a^{11}+18a^{10}+13a^{9}+14a^{8}+19a^{7}+18a^{6}+13a^{5}+7a^{4}+10a^{3}+12a^{2}+2a-1$, $45a^{24}-18a^{23}-17a^{22}+46a^{21}-34a^{20}+39a^{18}-38a^{17}+3a^{16}+56a^{15}-77a^{14}+50a^{13}+27a^{12}-90a^{11}+100a^{10}-37a^{9}-47a^{8}+89a^{7}-50a^{6}-48a^{5}+119a^{4}-100a^{3}-22a^{2}+157a-36$, $38a^{24}+28a^{23}+7a^{22}-14a^{21}-31a^{20}-49a^{19}-61a^{18}-64a^{17}-53a^{16}-31a^{15}-13a^{14}+17a^{13}+57a^{12}+76a^{11}+81a^{10}+91a^{9}+86a^{8}+42a^{7}-22a^{5}-75a^{4}-125a^{3}-127a^{2}-120a+39$ (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: | \( 101792268413442.38 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 101792268413442.38 \cdot 2}{2\cdot\sqrt{1501652986908498464865176954664165668592017173849}}\cr\approx \mathstrut & 0.628953323612443 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ are not computed |
Character table for $S_{25}$ 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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $16{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(38117394317861\) | $\Q_{38117394317861}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{38117394317861}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(393\!\cdots\!109\) | 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |