Normalized defining polynomial
\( x^{25} - 5x - 3 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-15898366049400615492894758682139527797698974609375\) \(\medspace = -\,3^{24}\cdot 5^{23}\cdot 409\cdot 11545399656835619219\) | 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: | \(92.91\) | 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: | \(3\), \(5\), \(409\), \(11545399656835619219\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-23610\!\cdots\!02855}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{5}a^{24}-\frac{2}{5}a^{23}-\frac{1}{5}a^{22}+\frac{2}{5}a^{21}+\frac{1}{5}a^{20}-\frac{2}{5}a^{19}-\frac{1}{5}a^{18}+\frac{2}{5}a^{17}+\frac{1}{5}a^{16}-\frac{2}{5}a^{15}-\frac{1}{5}a^{14}+\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$
Monogenic: | Not computed | |
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: | $a+1$, $a^{23}-a^{22}-a^{21}+a^{20}-a^{18}+a^{16}+a^{13}+a^{12}-a^{11}+a^{10}+3a^{9}-a^{8}-2a^{7}+4a^{6}-6a^{4}+2a^{3}+5a^{2}-7a-5$, $\frac{7}{5}a^{24}-\frac{19}{5}a^{23}+\frac{18}{5}a^{22}-\frac{6}{5}a^{21}-\frac{13}{5}a^{20}+\frac{26}{5}a^{19}-\frac{22}{5}a^{18}-\frac{1}{5}a^{17}+\frac{22}{5}a^{16}-\frac{29}{5}a^{15}+\frac{18}{5}a^{14}+\frac{4}{5}a^{13}-\frac{28}{5}a^{12}+\frac{31}{5}a^{11}-\frac{17}{5}a^{10}-\frac{6}{5}a^{9}+\frac{22}{5}a^{8}-\frac{34}{5}a^{7}+\frac{23}{5}a^{6}+\frac{4}{5}a^{5}-\frac{33}{5}a^{4}+\frac{41}{5}a^{3}-\frac{37}{5}a^{2}-\frac{6}{5}a+\frac{22}{5}$, $\frac{33}{5}a^{24}+\frac{14}{5}a^{23}-\frac{43}{5}a^{22}+\frac{6}{5}a^{21}+\frac{43}{5}a^{20}-\frac{31}{5}a^{19}-\frac{33}{5}a^{18}+\frac{56}{5}a^{17}+\frac{18}{5}a^{16}-\frac{71}{5}a^{15}+\frac{12}{5}a^{14}+\frac{71}{5}a^{13}-\frac{57}{5}a^{12}-\frac{51}{5}a^{11}+\frac{97}{5}a^{10}+\frac{6}{5}a^{9}-\frac{117}{5}a^{8}+\frac{44}{5}a^{7}+\frac{107}{5}a^{6}-\frac{89}{5}a^{5}-\frac{62}{5}a^{4}+\frac{144}{5}a^{3}-\frac{3}{5}a^{2}-\frac{184}{5}a-\frac{82}{5}$, $\frac{63}{5}a^{24}+\frac{69}{5}a^{23}+\frac{62}{5}a^{22}+\frac{26}{5}a^{21}-\frac{17}{5}a^{20}-\frac{56}{5}a^{19}-\frac{98}{5}a^{18}-\frac{124}{5}a^{17}-\frac{102}{5}a^{16}-\frac{51}{5}a^{15}+\frac{17}{5}a^{14}+\frac{96}{5}a^{13}+\frac{163}{5}a^{12}+\frac{194}{5}a^{11}+\frac{177}{5}a^{10}+\frac{96}{5}a^{9}-\frac{17}{5}a^{8}-\frac{141}{5}a^{7}-\frac{268}{5}a^{6}-\frac{334}{5}a^{5}-\frac{287}{5}a^{4}-\frac{176}{5}a^{3}-\frac{3}{5}a^{2}+\frac{231}{5}a+\frac{118}{5}$, $\frac{22}{5}a^{24}-\frac{9}{5}a^{23}+\frac{3}{5}a^{22}-\frac{16}{5}a^{21}+\frac{22}{5}a^{20}-\frac{14}{5}a^{19}-\frac{2}{5}a^{18}-\frac{21}{5}a^{17}+\frac{27}{5}a^{16}-\frac{9}{5}a^{15}+\frac{3}{5}a^{14}-\frac{16}{5}a^{13}+\frac{42}{5}a^{12}-\frac{4}{5}a^{11}-\frac{2}{5}a^{10}-\frac{31}{5}a^{9}+\frac{37}{5}a^{8}-\frac{14}{5}a^{7}-\frac{17}{5}a^{6}-\frac{46}{5}a^{5}+\frac{47}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{31}{5}a-\frac{28}{5}$, $15a^{24}+6a^{23}-22a^{22}+18a^{21}+8a^{20}-28a^{19}+18a^{18}+12a^{17}-33a^{16}+21a^{15}+18a^{14}-39a^{13}+22a^{12}+24a^{11}-51a^{10}+22a^{9}+32a^{8}-56a^{7}+26a^{6}+44a^{5}-69a^{4}+22a^{3}+54a^{2}-83a-56$, $\frac{4}{5}a^{24}-\frac{8}{5}a^{23}-\frac{14}{5}a^{22}+\frac{13}{5}a^{21}+\frac{19}{5}a^{20}-\frac{18}{5}a^{19}-\frac{9}{5}a^{18}+\frac{3}{5}a^{17}+\frac{4}{5}a^{16}+\frac{17}{5}a^{15}+\frac{1}{5}a^{14}-\frac{42}{5}a^{13}+\frac{9}{5}a^{12}+\frac{42}{5}a^{11}-\frac{4}{5}a^{10}-\frac{22}{5}a^{9}-\frac{11}{5}a^{8}-\frac{18}{5}a^{7}+\frac{51}{5}a^{6}+\frac{43}{5}a^{5}-\frac{66}{5}a^{4}-\frac{53}{5}a^{3}+\frac{51}{5}a^{2}+\frac{33}{5}a-\frac{1}{5}$, $\frac{33}{5}a^{24}-\frac{21}{5}a^{23}+\frac{17}{5}a^{22}-\frac{9}{5}a^{21}-\frac{7}{5}a^{20}-\frac{11}{5}a^{19}-\frac{3}{5}a^{18}-\frac{9}{5}a^{17}+\frac{23}{5}a^{16}-\frac{6}{5}a^{15}+\frac{12}{5}a^{14}+\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{9}{5}a^{11}+\frac{12}{5}a^{10}-\frac{19}{5}a^{9}+\frac{3}{5}a^{8}-\frac{46}{5}a^{7}-\frac{13}{5}a^{6}+\frac{6}{5}a^{5}+\frac{13}{5}a^{4}+\frac{29}{5}a^{3}+\frac{27}{5}a^{2}-\frac{19}{5}a-\frac{127}{5}$, $\frac{18}{5}a^{24}-\frac{6}{5}a^{23}+\frac{2}{5}a^{22}+\frac{1}{5}a^{21}+\frac{18}{5}a^{20}-\frac{16}{5}a^{19}+\frac{17}{5}a^{18}+\frac{11}{5}a^{17}-\frac{17}{5}a^{16}+\frac{19}{5}a^{15}-\frac{3}{5}a^{14}-\frac{9}{5}a^{13}-\frac{12}{5}a^{12}+\frac{19}{5}a^{11}-\frac{53}{5}a^{10}+\frac{11}{5}a^{9}-\frac{12}{5}a^{8}-\frac{41}{5}a^{7}-\frac{8}{5}a^{6}+\frac{1}{5}a^{5}-\frac{42}{5}a^{4}-\frac{16}{5}a^{3}+\frac{42}{5}a^{2}-\frac{69}{5}a-\frac{52}{5}$, $\frac{9}{5}a^{24}-\frac{23}{5}a^{23}-\frac{9}{5}a^{22}-\frac{12}{5}a^{21}-\frac{31}{5}a^{20}-\frac{3}{5}a^{19}-\frac{34}{5}a^{18}-\frac{7}{5}a^{17}-\frac{11}{5}a^{16}-\frac{18}{5}a^{15}+\frac{21}{5}a^{14}-\frac{17}{5}a^{13}+\frac{39}{5}a^{12}+\frac{12}{5}a^{11}+\frac{41}{5}a^{10}+\frac{58}{5}a^{9}+\frac{29}{5}a^{8}+\frac{82}{5}a^{7}+\frac{26}{5}a^{6}+\frac{63}{5}a^{5}+\frac{49}{5}a^{4}-\frac{3}{5}a^{3}+\frac{46}{5}a^{2}-\frac{72}{5}a-\frac{61}{5}$, $\frac{17}{5}a^{24}-\frac{19}{5}a^{23}+\frac{3}{5}a^{22}+\frac{19}{5}a^{21}-\frac{13}{5}a^{20}-\frac{4}{5}a^{19}+\frac{28}{5}a^{18}-\frac{26}{5}a^{17}+\frac{2}{5}a^{16}+\frac{11}{5}a^{15}-\frac{12}{5}a^{14}-\frac{21}{5}a^{13}+\frac{27}{5}a^{12}-\frac{9}{5}a^{11}-\frac{27}{5}a^{10}+\frac{54}{5}a^{9}-\frac{13}{5}a^{8}-\frac{24}{5}a^{7}+\frac{53}{5}a^{6}-\frac{6}{5}a^{5}-\frac{68}{5}a^{4}+\frac{81}{5}a^{3}-\frac{47}{5}a^{2}-\frac{66}{5}a+\frac{2}{5}$, $\frac{78}{5}a^{24}-\frac{6}{5}a^{23}-\frac{73}{5}a^{22}+\frac{101}{5}a^{21}-\frac{77}{5}a^{20}-\frac{1}{5}a^{19}+\frac{107}{5}a^{18}-\frac{149}{5}a^{17}+\frac{83}{5}a^{16}+\frac{29}{5}a^{15}-\frac{118}{5}a^{14}+\frac{166}{5}a^{13}-\frac{112}{5}a^{12}-\frac{51}{5}a^{11}+\frac{177}{5}a^{10}-\frac{184}{5}a^{9}+\frac{98}{5}a^{8}+\frac{64}{5}a^{7}-\frac{248}{5}a^{6}+\frac{266}{5}a^{5}-\frac{72}{5}a^{4}-\frac{136}{5}a^{3}+\frac{262}{5}a^{2}-\frac{294}{5}a-\frac{262}{5}$ (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: | \( 1137438749492350.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{11}\cdot 1137438749492350.0 \cdot 1}{2\cdot\sqrt{15898366049400615492894758682139527797698974609375}}\cr\approx \mathstrut & 0.687527884212649 \end{aligned}\] (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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | R | R | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $25$ | $20{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $18{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $[\ ]$ |
3.12.12.14 | $x^{12} + 54 x^{10} - 240 x^{9} - 1935 x^{8} + 7236 x^{7} + 45198 x^{6} + 60156 x^{5} + 16686 x^{4} - 13824 x^{3} + 5184 x^{2} - 972 x + 81$ | $3$ | $4$ | $12$ | 12T173 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
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 | $[3/2, 3/2]_{2}^{4}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $24$ | $24$ | $1$ | $23$ | ||||
\(409\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(11545399656835619219\) | $\Q_{11545399656835619219}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11545399656835619219}$ | $x$ | $1$ | $1$ | $0$ | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |