Normalized defining polynomial
\( x^{25} - 5x^{20} - 30x^{15} + 25x^{10} + 15x^{5} + 1 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2710505431213761085018632002174854278564453125\) \(\medspace = 5^{65}\) | 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: | \(65.66\) | 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: | \(5\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7}a^{15}+\frac{3}{7}a^{10}-\frac{1}{7}a^{5}-\frac{3}{7}$, $\frac{1}{7}a^{16}+\frac{3}{7}a^{11}-\frac{1}{7}a^{6}-\frac{3}{7}a$, $\frac{1}{7}a^{17}+\frac{3}{7}a^{12}-\frac{1}{7}a^{7}-\frac{3}{7}a^{2}$, $\frac{1}{7}a^{18}+\frac{3}{7}a^{13}-\frac{1}{7}a^{8}-\frac{3}{7}a^{3}$, $\frac{1}{7}a^{19}+\frac{3}{7}a^{14}-\frac{1}{7}a^{9}-\frac{3}{7}a^{4}$, $\frac{1}{49}a^{20}-\frac{1}{49}a^{15}+\frac{15}{49}a^{10}-\frac{13}{49}a^{5}+\frac{12}{49}$, $\frac{1}{49}a^{21}-\frac{1}{49}a^{16}+\frac{15}{49}a^{11}-\frac{13}{49}a^{6}+\frac{12}{49}a$, $\frac{1}{49}a^{22}-\frac{1}{49}a^{17}+\frac{15}{49}a^{12}-\frac{13}{49}a^{7}+\frac{12}{49}a^{2}$, $\frac{1}{49}a^{23}-\frac{1}{49}a^{18}+\frac{15}{49}a^{13}-\frac{13}{49}a^{8}+\frac{12}{49}a^{3}$, $\frac{1}{49}a^{24}-\frac{1}{49}a^{19}+\frac{15}{49}a^{14}-\frac{13}{49}a^{9}+\frac{12}{49}a^{4}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $14$ | 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: | $a^{24}+5a^{19}+30a^{14}-25a^{9}-15a^{4}$, $\frac{41}{49}a^{23}+\frac{209}{49}a^{18}+\frac{1212}{49}a^{13}-\frac{1154}{49}a^{8}-\frac{555}{49}a^{3}$, $\frac{8}{49}a^{22}+\frac{36}{49}a^{17}+\frac{258}{49}a^{12}-\frac{71}{49}a^{7}-\frac{131}{49}a^{2}$, $\frac{8}{49}a^{21}-\frac{43}{49}a^{16}-\frac{230}{49}a^{11}+\frac{323}{49}a^{6}+\frac{103}{49}a$, $\frac{87}{49}a^{24}-\frac{19}{49}a^{23}+\frac{8}{49}a^{22}+\frac{3}{49}a^{21}-\frac{18}{49}a^{20}+\frac{409}{49}a^{19}+\frac{82}{49}a^{18}-\frac{36}{49}a^{17}-\frac{3}{49}a^{16}+\frac{102}{49}a^{15}+\frac{2748}{49}a^{14}+\frac{639}{49}a^{13}-\frac{258}{49}a^{12}-\frac{151}{49}a^{11}+\frac{472}{49}a^{10}-\frac{1445}{49}a^{9}-\frac{110}{49}a^{8}+\frac{71}{49}a^{7}-\frac{284}{49}a^{6}-\frac{781}{49}a^{5}-\frac{2157}{49}a^{4}-\frac{711}{49}a^{3}+\frac{131}{49}a^{2}+\frac{330}{49}a+\frac{218}{49}$, $\frac{1}{7}a^{24}-\frac{4}{49}a^{23}-\frac{4}{49}a^{22}+\frac{9}{49}a^{21}-\frac{17}{49}a^{20}+\frac{3}{7}a^{19}+\frac{39}{49}a^{18}+\frac{39}{49}a^{17}-\frac{58}{49}a^{16}+\frac{66}{49}a^{15}+\frac{40}{7}a^{14}+\frac{45}{49}a^{13}+\frac{45}{49}a^{12}-\frac{208}{49}a^{11}+\frac{578}{49}a^{10}+\frac{32}{7}a^{9}-\frac{767}{49}a^{8}-\frac{767}{49}a^{7}+\frac{618}{49}a^{6}+\frac{270}{49}a^{5}-\frac{46}{7}a^{4}-\frac{104}{49}a^{3}-\frac{104}{49}a^{2}+\frac{10}{49}a+\frac{41}{49}$, $\frac{103}{49}a^{24}-\frac{95}{49}a^{23}-\frac{8}{49}a^{22}+\frac{33}{49}a^{21}+\frac{25}{49}a^{20}+\frac{523}{49}a^{19}+\frac{487}{49}a^{18}+\frac{43}{49}a^{17}-\frac{166}{49}a^{16}-\frac{130}{49}a^{15}+\frac{3047}{49}a^{14}+\frac{2789}{49}a^{13}+\frac{230}{49}a^{12}-\frac{982}{49}a^{11}-\frac{724}{49}a^{10}-\frac{2805}{49}a^{9}-\frac{2734}{49}a^{8}-\frac{323}{49}a^{7}+\frac{831}{49}a^{6}+\frac{760}{49}a^{5}-\frac{1222}{49}a^{4}-\frac{1042}{49}a^{3}-\frac{103}{49}a^{2}+\frac{452}{49}a+\frac{272}{49}$, $\frac{16}{49}a^{24}-\frac{53}{49}a^{23}+\frac{11}{49}a^{22}-\frac{11}{49}a^{21}-\frac{1}{49}a^{20}-\frac{79}{49}a^{19}+\frac{270}{49}a^{18}-\frac{53}{49}a^{17}+\frac{53}{49}a^{16}+\frac{8}{49}a^{15}-\frac{488}{49}a^{14}+\frac{1571}{49}a^{13}-\frac{353}{49}a^{12}+\frac{353}{49}a^{11}+\frac{6}{49}a^{10}+\frac{394}{49}a^{9}-\frac{1537}{49}a^{8}+\frac{340}{49}a^{7}-\frac{340}{49}a^{6}-\frac{43}{49}a^{5}+\frac{283}{49}a^{4}-\frac{601}{49}a^{3}+\frac{111}{49}a^{2}-\frac{111}{49}a+\frac{65}{49}$, $\frac{131}{7}a^{24}-\frac{228}{49}a^{23}+\frac{155}{49}a^{22}+\frac{188}{49}a^{21}+\frac{160}{49}a^{20}+95a^{19}+\frac{1173}{49}a^{18}-\frac{778}{49}a^{17}-\frac{944}{49}a^{16}-\frac{811}{49}a^{15}+\frac{3879}{7}a^{14}+\frac{6667}{49}a^{13}-\frac{4640}{49}a^{12}-\frac{5622}{49}a^{11}-\frac{4747}{49}a^{10}-510a^{9}-\frac{6654}{49}a^{8}+\frac{3998}{49}a^{7}+\frac{4829}{49}a^{6}+\frac{4353}{49}a^{5}-\frac{1690}{7}a^{4}-\frac{2337}{49}a^{3}+\frac{2357}{49}a^{2}+\frac{2858}{49}a+\frac{2109}{49}$, $\frac{8}{49}a^{24}+\frac{17}{49}a^{23}-\frac{17}{49}a^{22}+\frac{1}{49}a^{21}+\frac{3}{49}a^{20}-\frac{43}{49}a^{19}-\frac{87}{49}a^{18}+\frac{87}{49}a^{17}-\frac{1}{49}a^{16}-\frac{17}{49}a^{15}-\frac{230}{49}a^{14}-\frac{494}{49}a^{13}+\frac{494}{49}a^{12}-\frac{34}{49}a^{11}-\frac{95}{49}a^{10}+\frac{323}{49}a^{9}+\frac{437}{49}a^{8}-\frac{437}{49}a^{7}-\frac{209}{49}a^{6}+\frac{269}{49}a^{5}+\frac{54}{49}a^{4}+\frac{267}{49}a^{3}-\frac{267}{49}a^{2}-\frac{37}{49}a+\frac{78}{49}$, $\frac{608}{49}a^{24}+\frac{226}{49}a^{23}-\frac{349}{49}a^{22}+\frac{6}{49}a^{21}+\frac{93}{49}a^{20}-\frac{3086}{49}a^{19}-\frac{1143}{49}a^{18}+\frac{1777}{49}a^{17}-\frac{20}{49}a^{16}-\frac{464}{49}a^{15}-\frac{18012}{49}a^{14}-\frac{6711}{49}a^{13}+\frac{10319}{49}a^{12}-\frac{246}{49}a^{11}-\frac{2805}{49}a^{10}+\frac{16624}{49}a^{9}+\frac{6015}{49}a^{8}-\frac{9778}{49}a^{7}-\frac{15}{49}a^{6}+\frac{2396}{49}a^{5}+\frac{7723}{49}a^{4}+\frac{3013}{49}a^{3}-\frac{4258}{49}a^{2}+\frac{310}{49}a+\frac{1347}{49}$, $\frac{440}{49}a^{24}+\frac{20}{49}a^{23}+\frac{81}{49}a^{22}-\frac{19}{49}a^{21}-\frac{18}{7}a^{20}-\frac{2232}{49}a^{19}-\frac{62}{49}a^{18}-\frac{431}{49}a^{17}+\frac{89}{49}a^{16}+\frac{94}{7}a^{15}-\frac{13035}{49}a^{14}-\frac{806}{49}a^{13}-\frac{2285}{49}a^{12}+\frac{611}{49}a^{11}+74a^{10}+\frac{11948}{49}a^{9}-\frac{561}{49}a^{8}+\frac{2727}{49}a^{7}-\frac{313}{49}a^{6}-\frac{570}{7}a^{5}+\frac{5707}{49}a^{4}+\frac{1640}{49}a^{3}+\frac{160}{49}a^{2}-\frac{683}{49}a-\frac{136}{7}$, $\frac{141}{49}a^{24}-\frac{36}{49}a^{23}-\frac{66}{49}a^{22}+\frac{8}{49}a^{21}-\frac{9}{49}a^{20}+\frac{736}{49}a^{19}+\frac{183}{49}a^{18}+\frac{332}{49}a^{17}-\frac{57}{49}a^{16}+\frac{51}{49}a^{15}+\frac{4080}{49}a^{14}+\frac{1077}{49}a^{13}+\frac{1964}{49}a^{12}-\frac{174}{49}a^{11}+\frac{236}{49}a^{10}-\frac{4446}{49}a^{9}-\frac{1051}{49}a^{8}-\frac{1711}{49}a^{7}+\frac{827}{49}a^{6}-\frac{366}{49}a^{5}-\frac{1713}{49}a^{4}-\frac{775}{49}a^{3}-\frac{512}{49}a^{2}+\frac{47}{49}a+\frac{11}{49}$, $\frac{25}{49}a^{24}-\frac{214}{49}a^{23}-\frac{223}{49}a^{22}-\frac{132}{49}a^{21}-\frac{44}{49}a^{20}-\frac{123}{49}a^{19}+\frac{1082}{49}a^{18}+\frac{1126}{49}a^{17}+\frac{671}{49}a^{16}+\frac{233}{49}a^{15}-\frac{752}{49}a^{14}+\frac{6352}{49}a^{13}+\frac{6616}{49}a^{12}+\frac{3900}{49}a^{11}+\frac{1279}{49}a^{10}+\frac{557}{49}a^{9}-\frac{5681}{49}a^{8}-\frac{5893}{49}a^{7}-\frac{3674}{49}a^{6}-\frac{1479}{49}a^{5}+\frac{300}{49}a^{4}-\frac{2673}{49}a^{3}-\frac{2739}{49}a^{2}-\frac{1682}{49}a-\frac{654}{49}$ (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: | \( 12584077929327.963 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 12584077929327.963 \cdot 1}{2\cdot\sqrt{2710505431213761085018632002174854278564453125}}\cr\approx \mathstrut & 0.370864210287819 \end{aligned}\] (assuming GRH)
Galois group
$C_5^4:C_{20}$ (as 25T94):
A solvable group of order 12500 |
The 56 conjugacy class representatives for $C_5^4:C_{20}$ |
Character table for $C_5^4:C_{20}$ |
Intermediate fields
5.5.390625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 25 siblings: | data not computed |
Minimal sibling: | 25.5.2710505431213761085018632002174854278564453125.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $20{,}\,{\href{/padicField/2.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/3.5.0.1}{5} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.5.0.1}{5} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | $20{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | Deg $25$ | $25$ | $1$ | $65$ |