Normalized defining polynomial
\( x^{15} - 3 x^{14} + 7 x^{13} - 12 x^{12} + 8 x^{11} - 2 x^{10} - 17 x^{9} + 36 x^{8} - 49 x^{7} + \cdots + 3 \)
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: | \(98765528526263521\) \(\medspace = 13^{3}\cdot 31^{6}\cdot 37^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.58\) | 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: | $13^{1/2}31^{1/2}37^{1/2}\approx 122.11060560000512$ | ||
Ramified primes: | \(13\), \(31\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{481}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{12437511}a^{14}-\frac{1221304}{12437511}a^{13}-\frac{1764504}{4145837}a^{12}-\frac{5260330}{12437511}a^{11}-\frac{255860}{4145837}a^{10}-\frac{716396}{4145837}a^{9}+\frac{5569642}{12437511}a^{8}+\frac{2234315}{12437511}a^{7}+\frac{2186188}{12437511}a^{6}-\frac{4083305}{12437511}a^{5}-\frac{4076633}{12437511}a^{4}-\frac{2009043}{4145837}a^{3}+\frac{5081312}{12437511}a^{2}-\frac{1125285}{4145837}a+\frac{1278925}{4145837}$
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)
| |
Fundamental units: | $\frac{764082}{4145837}a^{14}-\frac{2390109}{4145837}a^{13}+\frac{5487053}{4145837}a^{12}-\frac{9120626}{4145837}a^{11}+\frac{5061319}{4145837}a^{10}+\frac{1878610}{4145837}a^{9}-\frac{17752671}{4145837}a^{8}+\frac{31259807}{4145837}a^{7}-\frac{34473809}{4145837}a^{6}+\frac{31970895}{4145837}a^{5}-\frac{15057718}{4145837}a^{4}+\frac{8125718}{4145837}a^{3}-\frac{2383}{4145837}a^{2}+\frac{1949528}{4145837}a-\frac{1835564}{4145837}$, $\frac{9923}{4145837}a^{14}+\frac{1991714}{12437511}a^{13}-\frac{3440195}{12437511}a^{12}+\frac{1979077}{4145837}a^{11}-\frac{6527150}{12437511}a^{10}-\frac{4352833}{4145837}a^{9}+\frac{3550356}{4145837}a^{8}-\frac{23214778}{12437511}a^{7}+\frac{15864694}{12437511}a^{6}+\frac{34295}{12437511}a^{5}-\frac{347113}{12437511}a^{4}+\frac{6076322}{12437511}a^{3}+\frac{4335219}{4145837}a^{2}-\frac{17321963}{12437511}a+\frac{1097154}{4145837}$, $\frac{43037}{12437511}a^{14}-\frac{338762}{12437511}a^{13}+\frac{337681}{4145837}a^{12}-\frac{1246988}{12437511}a^{11}-\frac{103748}{4145837}a^{10}+\frac{1055117}{4145837}a^{9}-\frac{7466749}{12437511}a^{8}+\frac{3817114}{12437511}a^{7}+\frac{9639752}{12437511}a^{6}-\frac{3604366}{12437511}a^{5}+\frac{9913256}{12437511}a^{4}-\frac{1752956}{4145837}a^{3}-\frac{16768880}{12437511}a^{2}+\frac{2777289}{4145837}a-\frac{3182624}{4145837}$, $\frac{4108882}{12437511}a^{14}-\frac{3576591}{4145837}a^{13}+\frac{21146300}{12437511}a^{12}-\frac{31822639}{12437511}a^{11}+\frac{102146}{12437511}a^{10}+\frac{7390772}{4145837}a^{9}-\frac{68229800}{12437511}a^{8}+\frac{35723543}{4145837}a^{7}-\frac{29238166}{4145837}a^{6}+\frac{93715400}{12437511}a^{5}-\frac{28473250}{12437511}a^{4}+\frac{22334392}{12437511}a^{3}-\frac{27364741}{12437511}a^{2}-\frac{1805479}{12437511}a-\frac{4277412}{4145837}$, $\frac{3084457}{12437511}a^{14}-\frac{7069433}{12437511}a^{13}+\frac{14339182}{12437511}a^{12}-\frac{18653359}{12437511}a^{11}-\frac{4368470}{12437511}a^{10}+\frac{5297332}{4145837}a^{9}-\frac{53001911}{12437511}a^{8}+\frac{52717714}{12437511}a^{7}-\frac{71479813}{12437511}a^{6}+\frac{11219023}{4145837}a^{5}-\frac{2746355}{4145837}a^{4}+\frac{18666161}{12437511}a^{3}+\frac{8630978}{12437511}a^{2}+\frac{12915139}{12437511}a+\frac{388203}{4145837}$, $\frac{3430810}{12437511}a^{14}-\frac{1826266}{4145837}a^{13}+\frac{12365630}{12437511}a^{12}-\frac{16055992}{12437511}a^{11}-\frac{10352422}{12437511}a^{10}-\frac{553680}{4145837}a^{9}-\frac{48742385}{12437511}a^{8}+\frac{14731605}{4145837}a^{7}-\frac{17882648}{4145837}a^{6}+\frac{54679562}{12437511}a^{5}-\frac{2163913}{12437511}a^{4}+\frac{32885212}{12437511}a^{3}+\frac{3604592}{12437511}a^{2}+\frac{9234086}{12437511}a-\frac{2055537}{4145837}$, $\frac{161350}{4145837}a^{14}-\frac{720022}{12437511}a^{13}-\frac{2363261}{12437511}a^{12}+\frac{2234325}{4145837}a^{11}-\frac{17916245}{12437511}a^{10}+\frac{7052065}{4145837}a^{9}+\frac{1816906}{4145837}a^{8}-\frac{16759951}{12437511}a^{7}+\frac{49157194}{12437511}a^{6}-\frac{62328892}{12437511}a^{5}+\frac{57874958}{12437511}a^{4}-\frac{35316409}{12437511}a^{3}+\frac{1403591}{4145837}a^{2}-\frac{9250874}{12437511}a+\frac{3119573}{4145837}$, $\frac{1325769}{4145837}a^{14}-\frac{14443930}{12437511}a^{13}+\frac{30955048}{12437511}a^{12}-\frac{17130013}{4145837}a^{11}+\frac{30872185}{12437511}a^{10}+\frac{7444440}{4145837}a^{9}-\frac{29388447}{4145837}a^{8}+\frac{170663729}{12437511}a^{7}-\frac{188437952}{12437511}a^{6}+\frac{174324848}{12437511}a^{5}-\frac{116794075}{12437511}a^{4}+\frac{46860392}{12437511}a^{3}-\frac{15820786}{4145837}a^{2}+\frac{12069151}{12437511}a-\frac{3456294}{4145837}$ | 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: | \( 363.728980494 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 363.728980494 \cdot 1}{2\cdot\sqrt{98765528526263521}}\cr\approx \mathstrut & 0.284848810825 \end{aligned}\]
Galois group
$S_3\times S_5$ (as 15T29):
A non-solvable group of order 720 |
The 21 conjugacy class representatives for $S_5 \times S_3$ |
Character table for $S_5 \times S_3$ |
Intermediate fields
3.1.31.1, 5.3.14911.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Degree 45 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 | $15$ | ${\href{/padicField/3.4.0.1}{4} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $15$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | R | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | R | R | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |