Normalized defining polynomial
\( x^{15} - 5 x^{14} + 16 x^{13} - 39 x^{12} + 65 x^{11} - 91 x^{10} + 78 x^{9} - 39 x^{8} - 26 x^{7} + 78 x^{6} - 78 x^{5} + 52 x^{4} - 13 x^{3} - 9 x^{2} + 6 x - 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1240576436601868288\) \(\medspace = 2^{12}\cdot 13^{13}\) | 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: | \(16.08\) | 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: | $2^{4/5}13^{11/12}\approx 18.278423529398573$ | ||
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: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}-\frac{4}{13}a^{11}+\frac{3}{13}a^{10}+\frac{1}{13}a^{9}-\frac{4}{13}a^{8}+\frac{3}{13}a^{7}+\frac{1}{13}a^{6}-\frac{4}{13}a^{5}+\frac{3}{13}a^{4}+\frac{1}{13}a^{3}-\frac{4}{13}a^{2}+\frac{3}{13}a+\frac{1}{13}$, $\frac{1}{13}a^{13}+\frac{4}{13}$, $\frac{1}{403}a^{14}+\frac{6}{403}a^{13}-\frac{11}{403}a^{12}+\frac{57}{403}a^{11}-\frac{176}{403}a^{10}-\frac{167}{403}a^{9}+\frac{70}{403}a^{8}-\frac{137}{403}a^{7}-\frac{76}{403}a^{6}-\frac{138}{403}a^{5}-\frac{46}{403}a^{4}+\frac{197}{403}a^{3}-\frac{47}{403}a^{2}-\frac{185}{403}a-\frac{13}{31}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{3}{13}a^{14}-\frac{4}{13}a^{13}+\frac{10}{13}a^{12}-\frac{14}{13}a^{11}-\frac{22}{13}a^{10}-\frac{29}{13}a^{9}-\frac{118}{13}a^{8}-\frac{74}{13}a^{7}-\frac{146}{13}a^{6}-\frac{40}{13}a^{5}+\frac{4}{13}a^{4}+\frac{10}{13}a^{3}+\frac{103}{13}a^{2}-\frac{10}{13}a-\frac{6}{13}$, $\frac{72}{403}a^{14}-\frac{219}{403}a^{13}+\frac{44}{31}a^{12}-\frac{73}{31}a^{11}+\frac{22}{31}a^{10}+\frac{17}{31}a^{9}-\frac{342}{31}a^{8}+\frac{269}{31}a^{7}-\frac{471}{31}a^{6}-\frac{37}{31}a^{5}+\frac{184}{31}a^{4}-\frac{292}{31}a^{3}+\frac{281}{31}a^{2}+\frac{444}{403}a-\frac{512}{403}$, $\frac{72}{403}a^{14}-\frac{219}{403}a^{13}+\frac{44}{31}a^{12}-\frac{73}{31}a^{11}+\frac{22}{31}a^{10}+\frac{17}{31}a^{9}-\frac{342}{31}a^{8}+\frac{269}{31}a^{7}-\frac{471}{31}a^{6}-\frac{37}{31}a^{5}+\frac{184}{31}a^{4}-\frac{292}{31}a^{3}+\frac{250}{31}a^{2}+\frac{847}{403}a-\frac{512}{403}$, $\frac{96}{403}a^{14}-\frac{32}{31}a^{13}+\frac{1145}{403}a^{12}-\frac{2526}{403}a^{11}+\frac{3409}{403}a^{10}-\frac{4159}{403}a^{9}+\frac{2349}{403}a^{8}-\frac{101}{403}a^{7}-\frac{1468}{403}a^{6}+\frac{3337}{403}a^{5}-\frac{2246}{403}a^{4}+\frac{1366}{403}a^{3}-\frac{1629}{403}a^{2}+\frac{127}{403}a+\frac{144}{403}$, $\frac{209}{403}a^{14}-\frac{978}{403}a^{13}+\frac{3002}{403}a^{12}-\frac{7276}{403}a^{11}+\frac{11762}{403}a^{10}-\frac{16706}{403}a^{9}+\frac{14382}{403}a^{8}-\frac{7088}{403}a^{7}-\frac{3732}{403}a^{6}+\frac{16449}{403}a^{5}-\frac{14667}{403}a^{4}+\frac{10607}{403}a^{3}-\frac{2414}{403}a^{2}-\frac{3821}{403}a+\frac{104}{31}$, $\frac{73}{403}a^{14}-\frac{492}{403}a^{13}+\frac{1584}{403}a^{12}-\frac{3775}{403}a^{11}+\frac{6403}{403}a^{10}-\frac{7386}{403}a^{9}+\frac{6040}{403}a^{8}+\frac{384}{403}a^{7}-\frac{4773}{403}a^{6}+\frac{6976}{403}a^{5}-\frac{6272}{403}a^{4}+\frac{245}{403}a^{3}-\frac{486}{403}a^{2}-\frac{54}{31}a+\frac{435}{403}$, $\frac{491}{403}a^{14}-\frac{2200}{403}a^{13}+\frac{6720}{403}a^{12}-\frac{15661}{403}a^{11}+\frac{23696}{403}a^{10}-\frac{31994}{403}a^{9}+\frac{20544}{403}a^{8}-\frac{6321}{403}a^{7}-\frac{19150}{403}a^{6}+\frac{31256}{403}a^{5}-\frac{23299}{403}a^{4}+\frac{12531}{403}a^{3}+\frac{2591}{403}a^{2}-\frac{5306}{403}a+\frac{96}{31}$, $\frac{824}{403}a^{14}-\frac{3612}{403}a^{13}+\frac{11117}{403}a^{12}-\frac{25696}{403}a^{11}+\frac{38434}{403}a^{10}-\frac{52141}{403}a^{9}+\frac{30555}{403}a^{8}-\frac{10836}{403}a^{7}-\frac{35189}{403}a^{6}+\frac{45752}{403}a^{5}-\frac{36199}{403}a^{4}+\frac{17279}{403}a^{3}+\frac{5478}{403}a^{2}-\frac{404}{31}a+\frac{1050}{403}$ | 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: | \( 1735.38854006 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 1735.38854006 \cdot 1}{2\cdot\sqrt{1240576436601868288}}\cr\approx \mathstrut & 0.383463615476 \end{aligned}\]
Galois group
$C_3\times F_5$ (as 15T8):
A solvable group of order 60 |
The 15 conjugacy class representatives for $F_5\times C_3$ |
Character table for $F_5\times C_3$ |
Intermediate fields
3.3.169.1, 5.1.35152.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 30 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $15$ | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | $15$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.15.12.1 | $x^{15} + 5 x^{13} + 5 x^{12} + 10 x^{11} + 26 x^{10} + 20 x^{9} - 145 x^{7} + 70 x^{6} + 73 x^{5} + 315 x^{4} - 105 x^{3} + 200 x^{2} + 5 x + 1$ | $5$ | $3$ | $12$ | $F_5\times C_3$ | $[\ ]_{5}^{12}$ |
\(13\) | 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.12.11.4 | $x^{12} + 13$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |