Normalized defining polynomial
\( x^{15} - 5 x^{14} + 8 x^{13} + x^{12} - 12 x^{11} + x^{10} + 22 x^{9} - 41 x^{8} + 44 x^{7} + 27 x^{6} + \cdots + 32 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-813831713247384370691\) \(\medspace = -\,971^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.78\) | 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: | $971^{1/2}\approx 31.160872901765767$ | ||
Ramified primes: | \(971\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-971}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{9}-\frac{1}{8}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{1376}a^{13}+\frac{27}{688}a^{12}-\frac{27}{1376}a^{11}-\frac{1}{16}a^{10}+\frac{9}{1376}a^{9}-\frac{35}{688}a^{8}+\frac{271}{1376}a^{7}+\frac{137}{688}a^{6}-\frac{121}{1376}a^{5}-\frac{41}{688}a^{4}+\frac{323}{1376}a^{3}-\frac{281}{688}a^{2}-\frac{117}{344}a-\frac{11}{43}$, $\frac{1}{527008}a^{14}-\frac{157}{527008}a^{13}-\frac{1789}{527008}a^{12}+\frac{6127}{527008}a^{11}-\frac{40841}{527008}a^{10}+\frac{48427}{527008}a^{9}+\frac{49785}{527008}a^{8}-\frac{105755}{527008}a^{7}-\frac{101967}{527008}a^{6}-\frac{30107}{527008}a^{5}+\frac{105517}{527008}a^{4}-\frac{34487}{527008}a^{3}+\frac{57853}{263504}a^{2}-\frac{33881}{131752}a+\frac{6191}{16469}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{455}{65876}a^{14}-\frac{241}{6128}a^{13}+\frac{10101}{131752}a^{12}-\frac{6033}{263504}a^{11}-\frac{5623}{65876}a^{10}+\frac{3095}{263504}a^{9}+\frac{27239}{131752}a^{8}-\frac{60645}{263504}a^{7}+\frac{9015}{131752}a^{6}+\frac{59669}{263504}a^{5}+\frac{14973}{65876}a^{4}-\frac{71155}{263504}a^{3}+\frac{45975}{131752}a^{2}+\frac{23709}{65876}a-\frac{8523}{16469}$, $\frac{1}{172}a^{14}-\frac{11}{344}a^{13}+\frac{13}{344}a^{12}+\frac{31}{344}a^{11}-\frac{17}{86}a^{10}-\frac{25}{172}a^{9}+\frac{93}{172}a^{8}-\frac{53}{344}a^{7}-\frac{125}{344}a^{6}+\frac{217}{344}a^{5}-\frac{11}{43}a^{4}-\frac{65}{172}a^{3}+\frac{19}{43}a^{2}-\frac{28}{43}a-\frac{53}{43}$, $\frac{368}{16469}a^{14}-\frac{52179}{527008}a^{13}+\frac{31423}{263504}a^{12}+\frac{58313}{527008}a^{11}-\frac{73815}{263504}a^{10}+\frac{18077}{527008}a^{9}+\frac{80481}{263504}a^{8}-\frac{376909}{527008}a^{7}+\frac{4015}{6128}a^{6}+\frac{12265}{12256}a^{5}+\frac{191763}{263504}a^{4}+\frac{1209631}{527008}a^{3}+\frac{660267}{263504}a^{2}+\frac{96067}{131752}a+\frac{29508}{16469}$, $\frac{1251}{131752}a^{14}-\frac{28437}{527008}a^{13}+\frac{26087}{263504}a^{12}+\frac{4571}{527008}a^{11}-\frac{59913}{263504}a^{10}+\frac{66271}{527008}a^{9}+\frac{102773}{263504}a^{8}-\frac{416347}{527008}a^{7}+\frac{128533}{263504}a^{6}+\frac{345769}{527008}a^{5}-\frac{143523}{263504}a^{4}+\frac{129029}{527008}a^{3}+\frac{375457}{263504}a^{2}-\frac{161455}{131752}a+\frac{5788}{16469}$, $\frac{2581}{527008}a^{14}-\frac{7087}{263504}a^{13}+\frac{29913}{527008}a^{12}-\frac{7227}{263504}a^{11}-\frac{41959}{527008}a^{10}+\frac{25637}{263504}a^{9}+\frac{68291}{527008}a^{8}-\frac{91485}{263504}a^{7}+\frac{95075}{527008}a^{6}+\frac{71015}{263504}a^{5}+\frac{89955}{527008}a^{4}+\frac{136815}{263504}a^{3}+\frac{119645}{131752}a^{2}-\frac{7937}{16469}a+\frac{977}{16469}$, $\frac{243}{32938}a^{14}+\frac{979}{131752}a^{13}-\frac{16557}{131752}a^{12}+\frac{6791}{65876}a^{11}+\frac{75183}{131752}a^{10}-\frac{97321}{131752}a^{9}-\frac{34919}{32938}a^{8}+\frac{223565}{131752}a^{7}+\frac{133717}{131752}a^{6}-\frac{134303}{65876}a^{5}+\frac{295013}{131752}a^{4}+\frac{539209}{131752}a^{3}+\frac{20563}{16469}a^{2}+\frac{86481}{32938}a+\frac{69913}{16469}$, $\frac{4843}{131752}a^{14}-\frac{2310}{16469}a^{13}+\frac{6993}{131752}a^{12}+\frac{28793}{65876}a^{11}-\frac{33211}{131752}a^{10}-\frac{63963}{65876}a^{9}+\frac{80727}{131752}a^{8}+\frac{34919}{32938}a^{7}-\frac{73305}{131752}a^{6}+\frac{48399}{65876}a^{5}+\frac{417019}{131752}a^{4}+\frac{152441}{65876}a^{3}+\frac{191461}{65876}a^{2}+\frac{84423}{16469}a+\frac{20497}{16469}$ | 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: | \( 178567.625718 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 178567.625718 \cdot 1}{2\cdot\sqrt{813831713247384370691}}\cr\approx \mathstrut & 2.41988372105 \end{aligned}\]
Galois group
A solvable group of order 30 |
The 9 conjugacy class representatives for $D_{15}$ |
Character table for $D_{15}$ |
Intermediate fields
3.1.971.1, 5.1.942841.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 30 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{7}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | $15$ | $15$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{5}$ | $15$ | $15$ | ${\href{/padicField/23.3.0.1}{3} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15$ | $15$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(971\) | $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |