Normalized defining polynomial
\( x^{15} - 3 x^{14} + 10 x^{13} - 18 x^{12} + 41 x^{11} - 61 x^{10} + 57 x^{9} - 103 x^{8} + 29 x^{7} + \cdots - 1 \)
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: | \(-6029359418000239616\) \(\medspace = -\,2^{10}\cdot 179^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.87\) | 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: | $2^{2/3}179^{1/2}\approx 21.237978619971454$ | ||
Ramified primes: | \(2\), \(179\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-179}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{14}a^{11}+\frac{1}{7}a^{10}-\frac{3}{14}a^{9}-\frac{3}{14}a^{8}-\frac{1}{2}a^{6}+\frac{3}{14}a^{5}+\frac{1}{7}a^{4}+\frac{5}{14}a^{3}+\frac{3}{14}a^{2}+\frac{3}{14}$, $\frac{1}{14}a^{12}+\frac{3}{14}a^{9}+\frac{3}{7}a^{8}+\frac{3}{14}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{2}a^{3}-\frac{3}{7}a^{2}-\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{196}a^{13}-\frac{1}{49}a^{12}+\frac{3}{196}a^{11}+\frac{23}{196}a^{10}-\frac{15}{196}a^{9}+\frac{93}{196}a^{8}+\frac{31}{196}a^{7}+\frac{19}{196}a^{6}+\frac{61}{196}a^{5}-\frac{33}{196}a^{4}-\frac{33}{196}a^{3}-\frac{27}{196}a^{2}-\frac{22}{49}a-\frac{9}{196}$, $\frac{1}{238924}a^{14}+\frac{225}{238924}a^{13}-\frac{3545}{238924}a^{12}+\frac{2113}{59731}a^{11}+\frac{19433}{119462}a^{10}-\frac{7195}{59731}a^{9}+\frac{14675}{119462}a^{8}+\frac{17827}{59731}a^{7}+\frac{51423}{119462}a^{6}-\frac{26135}{119462}a^{5}+\frac{28192}{59731}a^{4}+\frac{31047}{119462}a^{3}+\frac{62147}{238924}a^{2}-\frac{33055}{238924}a+\frac{38329}{238924}$
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)
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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: | $a$, $\frac{31103}{238924}a^{14}-\frac{37841}{119462}a^{13}+\frac{310479}{238924}a^{12}-\frac{539687}{238924}a^{11}+\frac{1472363}{238924}a^{10}-\frac{2089093}{238924}a^{9}+\frac{2768869}{238924}a^{8}-\frac{5252431}{238924}a^{7}+\frac{1785627}{238924}a^{6}-\frac{6233075}{238924}a^{5}+\frac{607561}{238924}a^{4}-\frac{2908093}{238924}a^{3}+\frac{81876}{59731}a^{2}-\frac{975389}{238924}a+\frac{158589}{119462}$, $\frac{81661}{238924}a^{14}-\frac{129345}{119462}a^{13}+\frac{840985}{238924}a^{12}-\frac{1543501}{238924}a^{11}+\frac{3381991}{238924}a^{10}-\frac{728559}{34132}a^{9}+\frac{648521}{34132}a^{8}-\frac{7624803}{238924}a^{7}+\frac{1985227}{238924}a^{6}-\frac{2618957}{238924}a^{5}+\frac{649601}{238924}a^{4}-\frac{1563077}{238924}a^{3}+\frac{265808}{59731}a^{2}+\frac{129699}{238924}a+\frac{3256}{59731}$, $\frac{379}{4876}a^{14}-\frac{1957}{119462}a^{13}+\frac{11309}{238924}a^{12}+\frac{241357}{238924}a^{11}-\frac{361209}{238924}a^{10}+\frac{1331975}{238924}a^{9}-\frac{2856051}{238924}a^{8}+\frac{2137125}{238924}a^{7}-\frac{5711529}{238924}a^{6}+\frac{2100933}{238924}a^{5}-\frac{2729995}{238924}a^{4}+\frac{703855}{238924}a^{3}-\frac{330703}{59731}a^{2}+\frac{448207}{238924}a+\frac{3259}{59731}$, $\frac{4253}{17066}a^{14}-\frac{62099}{119462}a^{13}+\frac{195427}{119462}a^{12}-\frac{104359}{59731}a^{11}+\frac{273061}{59731}a^{10}-\frac{200465}{59731}a^{9}-\frac{335400}{59731}a^{8}-\frac{284014}{59731}a^{7}-\frac{1311574}{59731}a^{6}+\frac{360278}{59731}a^{5}-\frac{390119}{59731}a^{4}+\frac{148300}{59731}a^{3}-\frac{427009}{119462}a^{2}+\frac{552217}{119462}a-\frac{105885}{119462}$, $\frac{5673}{59731}a^{14}-\frac{1026}{8533}a^{13}+\frac{57847}{119462}a^{12}-\frac{198}{1219}a^{11}+\frac{10302}{8533}a^{10}+\frac{22205}{59731}a^{9}-\frac{205452}{59731}a^{8}-\frac{158212}{59731}a^{7}-\frac{733574}{59731}a^{6}-\frac{263169}{59731}a^{5}-\frac{223851}{59731}a^{4}-\frac{32957}{59731}a^{3}+\frac{18044}{8533}a^{2}+\frac{108684}{59731}a+\frac{79381}{119462}$, $\frac{37671}{238924}a^{14}-\frac{31627}{59731}a^{13}+\frac{378083}{238924}a^{12}-\frac{674301}{238924}a^{11}+\frac{1314991}{238924}a^{10}-\frac{1878573}{238924}a^{9}+\frac{814171}{238924}a^{8}-\frac{155973}{34132}a^{7}-\frac{1656085}{238924}a^{6}+\frac{3589675}{238924}a^{5}-\frac{2419735}{238924}a^{4}+\frac{336141}{34132}a^{3}-\frac{294919}{119462}a^{2}+\frac{108795}{34132}a-\frac{111095}{119462}$ | 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: | \( 4800.51400574 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 4800.51400574 \cdot 1}{2\cdot\sqrt{6029359418000239616}}\cr\approx \mathstrut & 0.755807562105 \end{aligned}\]
Galois group
A solvable group of order 30 |
The 9 conjugacy class representatives for $D_{15}$ |
Character table for $D_{15}$ |
Intermediate fields
3.1.716.1, 5.1.32041.1 |
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 | R | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $15$ | $15$ | $15$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15$ | ${\href{/padicField/47.3.0.1}{3} }^{5}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(179\) | $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.179.2t1.a.a | $1$ | $ 179 $ | \(\Q(\sqrt{-179}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.716.3t2.a.a | $2$ | $ 2^{2} \cdot 179 $ | 3.1.716.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.179.5t2.a.b | $2$ | $ 179 $ | 5.1.32041.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.179.5t2.a.a | $2$ | $ 179 $ | 5.1.32041.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.716.15t2.a.b | $2$ | $ 2^{2} \cdot 179 $ | 15.1.6029359418000239616.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.716.15t2.a.a | $2$ | $ 2^{2} \cdot 179 $ | 15.1.6029359418000239616.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.716.15t2.a.c | $2$ | $ 2^{2} \cdot 179 $ | 15.1.6029359418000239616.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.716.15t2.a.d | $2$ | $ 2^{2} \cdot 179 $ | 15.1.6029359418000239616.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |