Normalized defining polynomial
\( x^{15} - 6 x^{14} + 24 x^{13} - 70 x^{12} + 126 x^{11} - 214 x^{10} + 284 x^{9} - 258 x^{8} + 349 x^{7} + \cdots - 256 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-72498065296628233355264\) \(\medspace = -\,2^{14}\cdot 461^{7}\) | 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: | \(33.42\) | 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: | $2\cdot 461^{1/2}\approx 42.941821107167776$ | ||
Ramified primes: | \(2\), \(461\) | 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: | \(\Q(\sqrt{-461}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}-\frac{1}{16}a^{6}+\frac{1}{16}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{9}+\frac{1}{16}a^{7}-\frac{3}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{12}-\frac{1}{64}a^{11}-\frac{1}{128}a^{10}-\frac{1}{64}a^{9}-\frac{5}{128}a^{8}-\frac{7}{64}a^{7}+\frac{13}{128}a^{6}-\frac{15}{64}a^{5}-\frac{3}{32}a^{4}+\frac{5}{16}a^{3}+\frac{1}{32}a^{2}-\frac{7}{16}a$, $\frac{1}{7424}a^{13}-\frac{25}{7424}a^{12}-\frac{51}{7424}a^{11}+\frac{149}{7424}a^{10}-\frac{407}{7424}a^{9}+\frac{389}{7424}a^{8}+\frac{751}{7424}a^{7}-\frac{73}{7424}a^{6}-\frac{685}{3712}a^{5}-\frac{265}{1856}a^{4}-\frac{325}{1856}a^{3}+\frac{427}{1856}a^{2}+\frac{321}{928}a-\frac{27}{58}$, $\frac{1}{29146624}a^{14}-\frac{385}{29146624}a^{13}+\frac{106679}{29146624}a^{12}+\frac{626697}{29146624}a^{11}-\frac{521121}{29146624}a^{10}-\frac{110883}{2242048}a^{9}-\frac{594531}{29146624}a^{8}-\frac{1258173}{29146624}a^{7}-\frac{323419}{3643328}a^{6}-\frac{241863}{1821664}a^{5}-\frac{161459}{7286656}a^{4}+\frac{1462679}{7286656}a^{3}+\frac{87315}{1821664}a^{2}-\frac{850131}{1821664}a-\frac{10017}{113854}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{87779}{14573312}a^{14}-\frac{154563}{3643328}a^{13}+\frac{1042595}{7286656}a^{12}-\frac{2916645}{7286656}a^{11}+\frac{490539}{910832}a^{10}-\frac{262255}{560512}a^{9}+\frac{7660533}{7286656}a^{8}+\frac{4914217}{7286656}a^{7}+\frac{26297177}{14573312}a^{6}-\frac{3672595}{7286656}a^{5}-\frac{192463}{455416}a^{4}-\frac{97443}{62816}a^{3}+\frac{2996753}{3643328}a^{2}+\frac{983669}{1821664}a+\frac{57769}{113854}$, $\frac{13261}{280256}a^{14}-\frac{353587}{1121024}a^{13}+\frac{1496529}{1121024}a^{12}-\frac{4612539}{1121024}a^{11}+\frac{9275523}{1121024}a^{10}-\frac{15808703}{1121024}a^{9}+\frac{21133427}{1121024}a^{8}-\frac{18309305}{1121024}a^{7}+\frac{14245853}{1121024}a^{6}-\frac{1107759}{560512}a^{5}-\frac{4609183}{280256}a^{4}+\frac{4257971}{280256}a^{3}-\frac{6147943}{280256}a^{2}+\frac{2328183}{140128}a-\frac{89701}{8758}$, $\frac{89649}{7286656}a^{14}-\frac{91697}{7286656}a^{13}+\frac{3723}{910832}a^{12}+\frac{1891727}{7286656}a^{11}-\frac{4921093}{3643328}a^{10}+\frac{784083}{560512}a^{9}-\frac{3656897}{910832}a^{8}+\frac{17142157}{7286656}a^{7}-\frac{7608563}{7286656}a^{6}+\frac{15912081}{3643328}a^{5}+\frac{614413}{910832}a^{4}+\frac{5351785}{1821664}a^{3}-\frac{8975003}{1821664}a^{2}+\frac{3245835}{910832}a-\frac{16550}{56927}$, $\frac{69533}{7286656}a^{14}+\frac{71083}{14573312}a^{13}-\frac{2143357}{14573312}a^{12}+\frac{9663463}{14573312}a^{11}-\frac{31969771}{14573312}a^{10}+\frac{2801295}{1121024}a^{9}-\frac{34131367}{14573312}a^{8}+\frac{83301469}{14573312}a^{7}-\frac{41247627}{14573312}a^{6}+\frac{755705}{7286656}a^{5}-\frac{27409169}{3643328}a^{4}+\frac{7633793}{3643328}a^{3}-\frac{13079303}{3643328}a^{2}+\frac{5544579}{1821664}a-\frac{580689}{113854}$, $\frac{307311}{7286656}a^{14}-\frac{3211103}{14573312}a^{13}+\frac{11848689}{14573312}a^{12}-\frac{31850251}{14573312}a^{11}+\frac{44008967}{14573312}a^{10}-\frac{5686339}{1121024}a^{9}+\frac{74691523}{14573312}a^{8}-\frac{19405481}{14573312}a^{7}+\frac{134298247}{14573312}a^{6}+\frac{20639955}{7286656}a^{5}+\frac{31896821}{3643328}a^{4}-\frac{14985085}{3643328}a^{3}+\frac{30471795}{3643328}a^{2}+\frac{1074865}{1821664}a+\frac{292921}{113854}$, $\frac{20389}{14573312}a^{14}-\frac{225473}{14573312}a^{13}+\frac{1134151}{14573312}a^{12}-\frac{3893951}{14573312}a^{11}+\frac{9494951}{14573312}a^{10}-\frac{1220427}{1121024}a^{9}+\frac{21044845}{14573312}a^{8}-\frac{21723477}{14573312}a^{7}+\frac{3755403}{3643328}a^{6}-\frac{2118369}{1821664}a^{5}+\frac{5467701}{3643328}a^{4}-\frac{3974401}{3643328}a^{3}+\frac{454245}{227708}a^{2}-\frac{1785881}{910832}a+\frac{70464}{56927}$, $\frac{823}{62816}a^{14}-\frac{1472}{56927}a^{13}+\frac{781335}{7286656}a^{12}+\frac{26365}{3643328}a^{11}-\frac{2763567}{7286656}a^{10}+\frac{208857}{280256}a^{9}-\frac{13217859}{7286656}a^{8}+\frac{6359835}{3643328}a^{7}-\frac{4329945}{7286656}a^{6}+\frac{7314347}{3643328}a^{5}-\frac{407177}{1821664}a^{4}+\frac{829003}{910832}a^{3}-\frac{3189077}{1821664}a^{2}+\frac{2497619}{910832}a-\frac{47160}{56927}$ (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)]
| |
Regulator: | \( 29224114.7546 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 29224114.7546 \cdot 1}{2\cdot\sqrt{72498065296628233355264}}\cr\approx \mathstrut & 41.9601695651 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 30 |
The 9 conjugacy class representatives for $D_{15}$ |
Character table for $D_{15}$ |
Intermediate fields
3.1.1844.1, 5.1.3400336.2 |
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 | R | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/7.3.0.1}{3} }^{5}$ | $15$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
\(461\) | $\Q_{461}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |