Normalized defining polynomial
\( x^{20} - 2 x^{19} + 10 x^{17} - 15 x^{16} + 40 x^{14} - 64 x^{13} + 46 x^{12} + 8 x^{11} - 32 x^{10} + \cdots + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(3354518684571451850752\) \(\medspace = 2^{16}\cdot 13^{15}\) | 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: | \(11.92\) | 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: | $2^{4/5}13^{3/4}\approx 11.920144243683046$ | ||
Ramified primes: | \(2\), \(13\) | 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: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{3}{8}a^{4}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{3}{8}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{8}+\frac{1}{8}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{9}+\frac{1}{8}a$, $\frac{1}{8}a^{18}-\frac{1}{4}a^{8}-\frac{1}{8}a^{2}+\frac{1}{4}$, $\frac{1}{16}a^{19}-\frac{1}{16}a^{18}-\frac{1}{16}a^{17}-\frac{1}{16}a^{16}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{2}a^{4}+\frac{1}{16}a^{3}+\frac{3}{16}a^{2}-\frac{1}{16}a-\frac{5}{16}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{9}{8}a^{19}-\frac{1}{4}a^{18}-\frac{21}{8}a^{17}+\frac{35}{4}a^{16}+\frac{9}{4}a^{15}-16a^{14}+\frac{55}{2}a^{13}+a^{12}-\frac{81}{4}a^{11}+27a^{10}+\frac{31}{2}a^{9}-26a^{8}+\frac{109}{4}a^{7}+12a^{6}-16a^{5}+9a^{4}+\frac{69}{8}a^{3}-\frac{19}{4}a^{2}+\frac{5}{8}a+\frac{9}{4}$, $\frac{5}{8}a^{19}-a^{18}-\frac{3}{4}a^{17}+\frac{13}{2}a^{16}-\frac{51}{8}a^{15}-\frac{49}{8}a^{14}+\frac{209}{8}a^{13}-\frac{211}{8}a^{12}+\frac{31}{8}a^{11}+\frac{173}{8}a^{10}-\frac{115}{8}a^{9}-\frac{97}{8}a^{8}+\frac{259}{8}a^{7}-\frac{171}{8}a^{6}-\frac{9}{8}a^{5}+\frac{95}{8}a^{4}-\frac{9}{2}a^{3}-\frac{17}{8}a^{2}+\frac{25}{8}a+\frac{1}{8}$, $\frac{1}{2}a^{19}-a^{18}+\frac{11}{2}a^{16}-8a^{15}-a^{14}+\frac{49}{2}a^{13}-34a^{12}+\frac{33}{2}a^{11}+\frac{39}{2}a^{10}-26a^{9}+\frac{3}{2}a^{8}+\frac{69}{2}a^{7}-34a^{6}+\frac{21}{2}a^{5}+\frac{31}{2}a^{4}-\frac{29}{2}a^{3}+\frac{5}{2}a^{2}+4a-\frac{3}{2}$, $a^{19}-\frac{1}{2}a^{18}-\frac{9}{4}a^{17}+\frac{67}{8}a^{16}-\frac{1}{8}a^{15}-\frac{115}{8}a^{14}+\frac{223}{8}a^{13}-\frac{49}{8}a^{12}-\frac{127}{8}a^{11}+\frac{219}{8}a^{10}+\frac{49}{8}a^{9}-\frac{185}{8}a^{8}+\frac{237}{8}a^{7}+\frac{15}{8}a^{6}-\frac{111}{8}a^{5}+\frac{97}{8}a^{4}+\frac{27}{8}a^{3}-\frac{35}{8}a^{2}+\frac{17}{8}a+\frac{3}{4}$, $\frac{3}{4}a^{19}-\frac{5}{2}a^{18}+\frac{1}{2}a^{17}+\frac{39}{4}a^{16}-\frac{157}{8}a^{15}+\frac{1}{8}a^{14}+\frac{355}{8}a^{13}-\frac{607}{8}a^{12}+\frac{325}{8}a^{11}+\frac{175}{8}a^{10}-\frac{411}{8}a^{9}-\frac{1}{8}a^{8}+\frac{461}{8}a^{7}-\frac{621}{8}a^{6}+\frac{225}{8}a^{5}+\frac{111}{8}a^{4}-\frac{187}{8}a^{3}+\frac{33}{8}a^{2}+\frac{35}{8}a-\frac{29}{8}$, $\frac{17}{16}a^{19}-\frac{33}{16}a^{18}-\frac{11}{16}a^{17}+\frac{173}{16}a^{16}-\frac{57}{4}a^{15}-\frac{21}{4}a^{14}+\frac{85}{2}a^{13}-\frac{119}{2}a^{12}+\frac{245}{8}a^{11}+\frac{133}{8}a^{10}-\frac{241}{8}a^{9}-\frac{47}{8}a^{8}+\frac{191}{4}a^{7}-\frac{229}{4}a^{6}+\frac{47}{2}a^{5}+4a^{4}-\frac{211}{16}a^{3}+\frac{79}{16}a^{2}+\frac{13}{16}a-\frac{23}{16}$, $a^{19}-a^{18}-a^{17}+\frac{17}{2}a^{16}-\frac{51}{8}a^{15}-\frac{21}{4}a^{14}+\frac{249}{8}a^{13}-\frac{135}{4}a^{12}+\frac{147}{8}a^{11}+16a^{10}-\frac{119}{8}a^{9}-\frac{11}{4}a^{8}+\frac{295}{8}a^{7}-\frac{131}{4}a^{6}+\frac{119}{8}a^{5}+\frac{23}{4}a^{4}-\frac{79}{8}a^{3}+3a^{2}+\frac{15}{8}a-\frac{7}{4}$, $\frac{11}{16}a^{19}-\frac{9}{16}a^{18}-\frac{17}{16}a^{17}+\frac{93}{16}a^{16}-\frac{5}{2}a^{15}-\frac{25}{4}a^{14}+20a^{13}-\frac{29}{2}a^{12}+\frac{31}{8}a^{11}+\frac{97}{8}a^{10}-\frac{13}{8}a^{9}-\frac{55}{8}a^{8}+22a^{7}-\frac{41}{4}a^{6}+4a^{5}+4a^{4}+\frac{15}{16}a^{3}+\frac{15}{16}a^{2}+\frac{11}{16}a+\frac{9}{16}$, $\frac{1}{8}a^{19}-\frac{1}{4}a^{18}+\frac{3}{8}a^{17}+a^{16}-\frac{19}{8}a^{15}+3a^{14}+\frac{31}{8}a^{13}-\frac{43}{4}a^{12}+\frac{127}{8}a^{11}-\frac{25}{4}a^{10}-\frac{5}{8}a^{9}+\frac{21}{4}a^{8}+\frac{59}{8}a^{7}-\frac{23}{2}a^{6}+\frac{125}{8}a^{5}-\frac{19}{4}a^{4}+2a^{3}+a^{2}+\frac{3}{4}a+\frac{1}{4}$ | 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: | \( 238.906276128 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 238.906276128 \cdot 1}{2\cdot\sqrt{3354518684571451850752}}\cr\approx \mathstrut & 0.197779438337 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 5 conjugacy class representatives for $F_5$ |
Character table for $F_5$ |
Intermediate fields
\(\Q(\sqrt{13}) \), 4.0.2197.1, 5.1.35152.1 x5, 10.2.16063620352.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.35152.1 |
Degree 10 sibling: | 10.2.16063620352.1 |
Minimal sibling: | 5.1.35152.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{5}$ | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{10}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $20$ | $5$ | $4$ | $16$ | |||
\(13\) | 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |