Normalized defining polynomial
\( x^{15} - 5 x^{14} + 10 x^{13} - 5 x^{12} - 15 x^{11} + 29 x^{10} - 8 x^{9} - 31 x^{8} + 36 x^{7} + \cdots + 1 \)
Invariants
Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(23713122135310336\) \(\medspace = 2^{18}\cdot 67^{6}\) | 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: | \(12.35\) | 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: | $2^{3/2}67^{1/2}\approx 23.15167380558045$ | ||
Ramified primes: | \(2\), \(67\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2474}a^{14}+\frac{204}{1237}a^{13}+\frac{141}{1237}a^{12}-\frac{527}{1237}a^{11}+\frac{107}{2474}a^{10}-\frac{156}{1237}a^{9}-\frac{108}{1237}a^{8}+\frac{531}{1237}a^{7}+\frac{372}{1237}a^{6}+\frac{248}{1237}a^{5}-\frac{259}{1237}a^{4}-\frac{579}{1237}a^{3}-\frac{763}{2474}a^{2}-\frac{465}{1237}a-\frac{310}{1237}$
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)
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Fundamental units: | $\frac{694}{1237}a^{14}-\frac{2595}{1237}a^{13}+\frac{6709}{2474}a^{12}+\frac{2065}{1237}a^{11}-\frac{9858}{1237}a^{10}+\frac{7369}{1237}a^{9}+\frac{18101}{2474}a^{8}-\frac{16305}{1237}a^{7}+\frac{1744}{1237}a^{6}+\frac{13945}{1237}a^{5}-\frac{6947}{1237}a^{4}-\frac{5787}{1237}a^{3}+\frac{8573}{1237}a^{2}+\frac{1531}{1237}a-\frac{3319}{2474}$, $\frac{694}{1237}a^{14}-\frac{2595}{1237}a^{13}+\frac{6709}{2474}a^{12}+\frac{2065}{1237}a^{11}-\frac{9858}{1237}a^{10}+\frac{7369}{1237}a^{9}+\frac{18101}{2474}a^{8}-\frac{16305}{1237}a^{7}+\frac{1744}{1237}a^{6}+\frac{13945}{1237}a^{5}-\frac{6947}{1237}a^{4}-\frac{5787}{1237}a^{3}+\frac{8573}{1237}a^{2}+\frac{1531}{1237}a-\frac{5793}{2474}$, $\frac{1169}{2474}a^{14}-\frac{4241}{2474}a^{13}+\frac{4327}{2474}a^{12}+\frac{3674}{1237}a^{11}-\frac{20883}{2474}a^{10}+\frac{10083}{2474}a^{9}+\frac{25821}{2474}a^{8}-\frac{17553}{1237}a^{7}-\frac{1793}{1237}a^{6}+\frac{17772}{1237}a^{5}-\frac{7128}{1237}a^{4}-\frac{6397}{1237}a^{3}+\frac{13537}{2474}a^{2}+\frac{5101}{2474}a-\frac{3609}{2474}$, $\frac{1169}{2474}a^{14}-\frac{4241}{2474}a^{13}+\frac{4327}{2474}a^{12}+\frac{3674}{1237}a^{11}-\frac{20883}{2474}a^{10}+\frac{10083}{2474}a^{9}+\frac{25821}{2474}a^{8}-\frac{17553}{1237}a^{7}-\frac{1793}{1237}a^{6}+\frac{17772}{1237}a^{5}-\frac{7128}{1237}a^{4}-\frac{6397}{1237}a^{3}+\frac{13537}{2474}a^{2}+\frac{5101}{2474}a-\frac{6083}{2474}$, $\frac{73}{1237}a^{14}+\frac{96}{1237}a^{13}-\frac{2123}{2474}a^{12}+\frac{2226}{1237}a^{11}-\frac{848}{1237}a^{10}-\frac{2984}{1237}a^{9}+\frac{9285}{2474}a^{8}-\frac{405}{1237}a^{7}-\frac{3827}{1237}a^{6}+\frac{2809}{1237}a^{5}-\frac{704}{1237}a^{4}+\frac{819}{1237}a^{3}-\frac{34}{1237}a^{2}+\frac{145}{1237}a+\frac{2255}{2474}$, $\frac{802}{1237}a^{14}-\frac{7363}{2474}a^{13}+\frac{13193}{2474}a^{12}-\frac{1674}{1237}a^{11}-\frac{11909}{1237}a^{10}+\frac{35173}{2474}a^{9}+\frac{1133}{2474}a^{8}-\frac{22835}{1237}a^{7}+\frac{17772}{1237}a^{6}+\frac{6900}{1237}a^{5}-\frac{13411}{1237}a^{4}+\frac{1508}{1237}a^{3}+\frac{7811}{1237}a^{2}-\frac{6083}{2474}a-\frac{1169}{2474}$, $\frac{581}{2474}a^{14}-\frac{1465}{1237}a^{13}+\frac{2753}{1237}a^{12}-\frac{648}{1237}a^{11}-\frac{12053}{2474}a^{10}+\frac{9561}{1237}a^{9}-\frac{898}{1237}a^{8}-\frac{13109}{1237}a^{7}+\frac{13264}{1237}a^{6}+\frac{1833}{1237}a^{5}-\frac{11935}{1237}a^{4}+\frac{6250}{1237}a^{3}+\frac{6965}{2474}a^{2}-\frac{4210}{1237}a-\frac{745}{1237}$, $\frac{2773}{2474}a^{14}-\frac{5802}{1237}a^{13}+\frac{8760}{1237}a^{12}+\frac{2000}{1237}a^{11}-\frac{44701}{2474}a^{10}+\frac{22628}{1237}a^{9}+\frac{13477}{1237}a^{8}-\frac{40388}{1237}a^{7}+\frac{15979}{1237}a^{6}+\frac{24672}{1237}a^{5}-\frac{21776}{1237}a^{4}-\frac{3652}{1237}a^{3}+\frac{29159}{2474}a^{2}-\frac{1728}{1237}a-\frac{3626}{1237}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 133.707274982 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 133.707274982 \cdot 1}{2\cdot\sqrt{23713122135310336}}\cr\approx \mathstrut & 0.213697710939 \end{aligned}\]
Galois group
A non-solvable group of order 60 |
The 5 conjugacy class representatives for $A_5$ |
Character table for $A_5$ |
Intermediate fields
5.1.287296.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.287296.2 |
Degree 6 sibling: | 6.2.287296.2 |
Degree 10 sibling: | 10.2.82538991616.1 |
Degree 12 sibling: | deg 12 |
Degree 20 sibling: | deg 20 |
Degree 30 sibling: | data not computed |
Minimal sibling: | 5.1.287296.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{5}$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{5}$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{5}$ | ${\href{/padicField/29.3.0.1}{3} }^{5}$ | ${\href{/padicField/31.3.0.1}{3} }^{5}$ | ${\href{/padicField/37.5.0.1}{5} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.5.0.1}{5} }^{3}$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.12.18.59 | $x^{12} + 6 x^{11} + 22 x^{10} + 56 x^{9} + 126 x^{8} + 240 x^{7} + 332 x^{6} - 18 x^{5} - 459 x^{4} - 394 x^{3} - 344 x^{2} + 138 x + 423$ | $4$ | $3$ | $18$ | $A_4$ | $[2, 2]^{3}$ | |
\(67\) | $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |