Normalized defining polynomial
\( x^{15} + 15 x^{13} - 7 x^{12} + 99 x^{11} - 48 x^{10} + 306 x^{9} + 360 x^{8} + 651 x^{7} - 292 x^{6} + \cdots + 603 \)
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)
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Discriminant: | \(-428830500916100995345287\) \(\medspace = -\,3^{20}\cdot 103^{7}\) | 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: | \(37.63\) | 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: | $3^{4/3}103^{1/2}\approx 43.91170349655221$ | ||
Ramified primes: | \(3\), \(103\) | 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: | \(\Q(\sqrt{-103}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5}a^{13}-\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{81\!\cdots\!35}a^{14}-\frac{25\!\cdots\!49}{27\!\cdots\!45}a^{13}+\frac{58\!\cdots\!24}{54\!\cdots\!69}a^{12}+\frac{35\!\cdots\!77}{81\!\cdots\!35}a^{11}-\frac{13\!\cdots\!07}{27\!\cdots\!45}a^{10}-\frac{20\!\cdots\!72}{54\!\cdots\!69}a^{9}-\frac{95\!\cdots\!19}{25\!\cdots\!35}a^{8}+\frac{30\!\cdots\!51}{27\!\cdots\!45}a^{7}+\frac{97\!\cdots\!02}{27\!\cdots\!45}a^{6}-\frac{67\!\cdots\!72}{16\!\cdots\!07}a^{5}-\frac{59\!\cdots\!47}{27\!\cdots\!45}a^{4}+\frac{22\!\cdots\!41}{75\!\cdots\!55}a^{3}+\frac{60\!\cdots\!91}{16\!\cdots\!07}a^{2}-\frac{76\!\cdots\!54}{27\!\cdots\!45}a+\frac{99\!\cdots\!57}{27\!\cdots\!45}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $\frac{53\!\cdots\!66}{27\!\cdots\!45}a^{14}-\frac{16\!\cdots\!39}{27\!\cdots\!45}a^{13}+\frac{82\!\cdots\!82}{27\!\cdots\!45}a^{12}-\frac{61\!\cdots\!81}{27\!\cdots\!45}a^{11}+\frac{58\!\cdots\!77}{27\!\cdots\!45}a^{10}-\frac{84\!\cdots\!08}{54\!\cdots\!69}a^{9}+\frac{18\!\cdots\!08}{25\!\cdots\!35}a^{8}+\frac{13\!\cdots\!51}{27\!\cdots\!45}a^{7}+\frac{37\!\cdots\!38}{27\!\cdots\!45}a^{6}-\frac{29\!\cdots\!02}{54\!\cdots\!69}a^{5}+\frac{37\!\cdots\!44}{27\!\cdots\!45}a^{4}+\frac{37\!\cdots\!51}{75\!\cdots\!55}a^{3}-\frac{60\!\cdots\!07}{27\!\cdots\!45}a^{2}+\frac{98\!\cdots\!76}{27\!\cdots\!45}a+\frac{14\!\cdots\!03}{27\!\cdots\!45}$, $\frac{43\!\cdots\!37}{27\!\cdots\!45}a^{14}-\frac{55\!\cdots\!43}{54\!\cdots\!69}a^{13}+\frac{70\!\cdots\!16}{27\!\cdots\!45}a^{12}-\frac{70\!\cdots\!67}{54\!\cdots\!69}a^{11}+\frac{51\!\cdots\!87}{27\!\cdots\!45}a^{10}-\frac{57\!\cdots\!36}{54\!\cdots\!69}a^{9}+\frac{17\!\cdots\!86}{25\!\cdots\!35}a^{8}+\frac{21\!\cdots\!22}{54\!\cdots\!69}a^{7}+\frac{45\!\cdots\!78}{27\!\cdots\!45}a^{6}+\frac{69\!\cdots\!14}{54\!\cdots\!69}a^{5}+\frac{49\!\cdots\!53}{27\!\cdots\!45}a^{4}-\frac{86\!\cdots\!99}{15\!\cdots\!91}a^{3}-\frac{28\!\cdots\!86}{27\!\cdots\!45}a^{2}-\frac{28\!\cdots\!37}{54\!\cdots\!69}a-\frac{21\!\cdots\!47}{27\!\cdots\!45}$, $\frac{22\!\cdots\!97}{27\!\cdots\!45}a^{14}-\frac{41\!\cdots\!64}{27\!\cdots\!45}a^{13}+\frac{85\!\cdots\!70}{54\!\cdots\!69}a^{12}-\frac{75\!\cdots\!46}{27\!\cdots\!45}a^{11}+\frac{38\!\cdots\!48}{27\!\cdots\!45}a^{10}-\frac{11\!\cdots\!28}{54\!\cdots\!69}a^{9}+\frac{15\!\cdots\!66}{25\!\cdots\!35}a^{8}-\frac{82\!\cdots\!39}{27\!\cdots\!45}a^{7}+\frac{20\!\cdots\!87}{27\!\cdots\!45}a^{6}+\frac{11\!\cdots\!06}{54\!\cdots\!69}a^{5}+\frac{63\!\cdots\!08}{27\!\cdots\!45}a^{4}-\frac{15\!\cdots\!34}{75\!\cdots\!55}a^{3}-\frac{19\!\cdots\!46}{54\!\cdots\!69}a^{2}+\frac{59\!\cdots\!76}{27\!\cdots\!45}a+\frac{13\!\cdots\!47}{27\!\cdots\!45}$, $\frac{16\!\cdots\!51}{27\!\cdots\!45}a^{14}-\frac{20\!\cdots\!66}{27\!\cdots\!45}a^{13}+\frac{19\!\cdots\!34}{27\!\cdots\!45}a^{12}-\frac{44\!\cdots\!39}{27\!\cdots\!45}a^{11}+\frac{22\!\cdots\!87}{54\!\cdots\!69}a^{10}-\frac{55\!\cdots\!12}{54\!\cdots\!69}a^{9}+\frac{21\!\cdots\!88}{25\!\cdots\!35}a^{8}+\frac{25\!\cdots\!39}{27\!\cdots\!45}a^{7}-\frac{10\!\cdots\!36}{54\!\cdots\!69}a^{6}-\frac{78\!\cdots\!07}{54\!\cdots\!69}a^{5}-\frac{18\!\cdots\!86}{27\!\cdots\!45}a^{4}+\frac{99\!\cdots\!84}{75\!\cdots\!55}a^{3}+\frac{39\!\cdots\!51}{27\!\cdots\!45}a^{2}-\frac{26\!\cdots\!36}{27\!\cdots\!45}a-\frac{44\!\cdots\!90}{54\!\cdots\!69}$, $\frac{60\!\cdots\!10}{54\!\cdots\!69}a^{14}+\frac{79\!\cdots\!78}{54\!\cdots\!69}a^{13}+\frac{95\!\cdots\!57}{54\!\cdots\!69}a^{12}+\frac{64\!\cdots\!84}{54\!\cdots\!69}a^{11}+\frac{58\!\cdots\!56}{54\!\cdots\!69}a^{10}+\frac{23\!\cdots\!33}{54\!\cdots\!69}a^{9}+\frac{16\!\cdots\!46}{50\!\cdots\!67}a^{8}+\frac{31\!\cdots\!85}{54\!\cdots\!69}a^{7}+\frac{78\!\cdots\!62}{54\!\cdots\!69}a^{6}+\frac{66\!\cdots\!95}{54\!\cdots\!69}a^{5}-\frac{73\!\cdots\!22}{54\!\cdots\!69}a^{4}-\frac{22\!\cdots\!56}{15\!\cdots\!91}a^{3}-\frac{23\!\cdots\!98}{54\!\cdots\!69}a^{2}+\frac{47\!\cdots\!89}{54\!\cdots\!69}a+\frac{23\!\cdots\!86}{54\!\cdots\!69}$, $\frac{13\!\cdots\!14}{27\!\cdots\!45}a^{14}-\frac{44\!\cdots\!01}{27\!\cdots\!45}a^{13}+\frac{24\!\cdots\!68}{27\!\cdots\!45}a^{12}-\frac{56\!\cdots\!14}{27\!\cdots\!45}a^{11}+\frac{20\!\cdots\!68}{27\!\cdots\!45}a^{10}-\frac{46\!\cdots\!01}{54\!\cdots\!69}a^{9}+\frac{50\!\cdots\!37}{25\!\cdots\!35}a^{8}+\frac{11\!\cdots\!89}{27\!\cdots\!45}a^{7}-\frac{28\!\cdots\!38}{27\!\cdots\!45}a^{6}+\frac{11\!\cdots\!73}{54\!\cdots\!69}a^{5}+\frac{59\!\cdots\!66}{27\!\cdots\!45}a^{4}-\frac{13\!\cdots\!76}{75\!\cdots\!55}a^{3}-\frac{49\!\cdots\!68}{27\!\cdots\!45}a^{2}+\frac{23\!\cdots\!69}{27\!\cdots\!45}a+\frac{24\!\cdots\!77}{27\!\cdots\!45}$, $\frac{62\!\cdots\!68}{27\!\cdots\!45}a^{14}+\frac{50\!\cdots\!24}{27\!\cdots\!45}a^{13}+\frac{19\!\cdots\!39}{54\!\cdots\!69}a^{12}+\frac{29\!\cdots\!56}{27\!\cdots\!45}a^{11}+\frac{62\!\cdots\!02}{27\!\cdots\!45}a^{10}+\frac{23\!\cdots\!55}{54\!\cdots\!69}a^{9}+\frac{17\!\cdots\!44}{25\!\cdots\!35}a^{8}+\frac{33\!\cdots\!49}{27\!\cdots\!45}a^{7}+\frac{66\!\cdots\!58}{27\!\cdots\!45}a^{6}+\frac{79\!\cdots\!09}{54\!\cdots\!69}a^{5}+\frac{43\!\cdots\!87}{27\!\cdots\!45}a^{4}+\frac{19\!\cdots\!44}{75\!\cdots\!55}a^{3}-\frac{62\!\cdots\!42}{54\!\cdots\!69}a^{2}-\frac{58\!\cdots\!86}{27\!\cdots\!45}a-\frac{44\!\cdots\!57}{27\!\cdots\!45}$ (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: | \( 1012274.34074 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 1012274.34074 \cdot 1}{2\cdot\sqrt{428830500916100995345287}}\cr\approx \mathstrut & 0.597605345152 \end{aligned}\] (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
3.1.8343.1, 5.1.10609.1 |
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 | $15$ | R | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $15$ | $15$ | $15$ | $15$ | $15$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ |
3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ | |
3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ | |
\(103\) | $\Q_{103}$ | $x + 98$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |