Properties

Label 20.0.433...000.1
Degree $20$
Signature $(0, 10)$
Discriminant $4.336\times 10^{51}$
Root discriminant \(381.82\)
Ramified primes $2,3,5,7$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $A_5^4.D_4^2.C_2$ (as 20T1103)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 + 175*x^12 - 280*x^11 + 112*x^10 + 4375*x^4 - 14000*x^3 + 16800*x^2 - 8960*x + 1792)
 
Copy content gp:K = bnfinit(y^20 + 175*y^12 - 280*y^11 + 112*y^10 + 4375*y^4 - 14000*y^3 + 16800*y^2 - 8960*y + 1792, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 175*x^12 - 280*x^11 + 112*x^10 + 4375*x^4 - 14000*x^3 + 16800*x^2 - 8960*x + 1792);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 175*x^12 - 280*x^11 + 112*x^10 + 4375*x^4 - 14000*x^3 + 16800*x^2 - 8960*x + 1792)
 

\( x^{20} + 175x^{12} - 280x^{11} + 112x^{10} + 4375x^{4} - 14000x^{3} + 16800x^{2} - 8960x + 1792 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 10)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(4336370964135621995388739780608000000000000000000000\) \(\medspace = 2^{52}\cdot 3^{11}\cdot 5^{21}\cdot 7^{19}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(381.82\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(3\), \(5\), \(7\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{105}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-14 +2 \sqrt{21}})\)

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1024}a^{17}+\frac{103}{256}a^{16}-\frac{5}{32}a^{15}+\frac{1}{8}a^{14}-\frac{1}{4}a^{13}+\frac{175}{1024}a^{9}+\frac{35}{256}a^{8}+\frac{7}{64}a^{7}-\frac{5}{16}a^{6}-\frac{1}{4}a^{5}+\frac{279}{1024}a-\frac{107}{256}$, $\frac{1}{1048576}a^{18}-\frac{51}{131072}a^{17}+\frac{7803}{65536}a^{16}-\frac{1579}{4096}a^{15}+\frac{1237}{4096}a^{14}-\frac{169}{512}a^{13}-\frac{113}{256}a^{12}-\frac{3}{8}a^{11}-\frac{458577}{1048576}a^{10}+\frac{221}{512}a^{9}-\frac{7}{128}a^{8}+\frac{5}{32}a^{7}+\frac{1}{8}a^{6}-\frac{1}{2}a^{5}+\frac{4375}{1048576}a^{2}+\frac{37269}{131072}a+\frac{24239}{65536}$, $\frac{1}{1073741824}a^{19}-\frac{51}{268435456}a^{18}+\frac{2601}{67108864}a^{17}-\frac{132651}{16777216}a^{16}-\frac{1623407}{4194304}a^{15}-\frac{43747}{1048576}a^{14}-\frac{128199}{262144}a^{13}-\frac{15451}{65536}a^{12}+\frac{102826159}{1073741824}a^{11}+\frac{124574941}{268435456}a^{10}+\frac{21}{64}a^{9}+\frac{1}{16}a^{8}+\frac{1}{4}a^{7}+\frac{4375}{1073741824}a^{3}-\frac{226625}{268435456}a^{2}+\frac{11558925}{67108864}a-\frac{2302755}{16777216}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{4}$, which has order $4$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{4}$, which has order $4$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{9765625}{2097152}a^{19}+\frac{3515625}{1048576}a^{18}+\frac{78125}{32768}a^{17}+\frac{109375}{65536}a^{16}+\frac{9375}{8192}a^{15}+\frac{3125}{4096}a^{14}+\frac{125}{256}a^{13}+\frac{75}{256}a^{12}+\frac{1709312055}{2097152}a^{11}-\frac{751887589}{1048576}a^{10}+\frac{42724609375}{2097152}a^{3}-\frac{52978515625}{1048576}a^{2}+\frac{341796875}{8192}a-\frac{751559909}{65536}$, $\frac{30545770942413}{67108864}a^{19}-\frac{133317954912807}{16777216}a^{18}-\frac{21643107221067}{4194304}a^{17}-\frac{17729916035039}{1048576}a^{16}-\frac{13468019914019}{262144}a^{15}-\frac{1096989938519}{65536}a^{14}+\frac{52546553157}{16384}a^{13}-\frac{187385002639}{4096}a^{12}+\frac{21\cdots 19}{67108864}a^{11}-\frac{16\cdots 27}{16777216}a^{10}+\frac{1713044069967}{1024}a^{9}-\frac{326418362301}{256}a^{8}-\frac{216286629273}{64}a^{7}+\frac{124427562075}{16}a^{6}-\frac{9989779333}{4}a^{5}-14610478393a^{4}+\frac{24\cdots 55}{67108864}a^{3}-\frac{71\cdots 97}{16777216}a^{2}+\frac{96\cdots 61}{4194304}a-\frac{49\cdots 27}{1048576}$, $\frac{43\cdots 99}{536870912}a^{19}-\frac{58\cdots 85}{134217728}a^{18}+\frac{14\cdots 11}{33554432}a^{17}+\frac{42\cdots 51}{8388608}a^{16}-\frac{40\cdots 65}{2097152}a^{15}+\frac{80\cdots 39}{524288}a^{14}+\frac{36\cdots 35}{131072}a^{13}-\frac{27\cdots 29}{32768}a^{12}+\frac{10\cdots 21}{536870912}a^{11}-\frac{11\cdots 93}{134217728}a^{10}+\frac{87\cdots 35}{512}a^{9}-\frac{856192317629793}{128}a^{8}-\frac{11\cdots 77}{32}a^{7}+\frac{569462929114167}{8}a^{6}-\frac{24620061379949}{2}a^{5}-171762113877783a^{4}+\frac{17\cdots 61}{536870912}a^{3}-\frac{37\cdots 03}{134217728}a^{2}+\frac{40\cdots 59}{33554432}a-\frac{17\cdots 01}{8388608}$, $\frac{68\cdots 65}{536870912}a^{19}-\frac{38\cdots 39}{134217728}a^{18}+\frac{18\cdots 65}{33554432}a^{17}-\frac{77\cdots 79}{8388608}a^{16}+\frac{29\cdots 97}{2097152}a^{15}-\frac{972999061854907}{524288}a^{14}+\frac{240920671075649}{131072}a^{13}-\frac{13228615209459}{32768}a^{12}+\frac{95\cdots 11}{536870912}a^{11}-\frac{92\cdots 43}{134217728}a^{10}+\frac{4691906075245}{32}a^{9}-\frac{2035014071751}{8}a^{8}+\frac{774374419335}{2}a^{7}-500811750263a^{6}+481919173613a^{5}-95857346515a^{4}-\frac{28\cdots 05}{536870912}a^{3}+\frac{97\cdots 51}{134217728}a^{2}-\frac{13\cdots 43}{33554432}a+\frac{64\cdots 73}{8388608}$, $\frac{18\cdots 85}{1073741824}a^{19}-\frac{12\cdots 23}{268435456}a^{18}+\frac{11\cdots 53}{67108864}a^{17}+\frac{25\cdots 49}{16777216}a^{16}-\frac{14\cdots 99}{4194304}a^{15}+\frac{627468295303333}{1048576}a^{14}+\frac{33\cdots 77}{262144}a^{13}-\frac{15\cdots 31}{65536}a^{12}+\frac{22\cdots 51}{1073741824}a^{11}-\frac{71\cdots 35}{268435456}a^{10}+\frac{322112869013961}{1024}a^{9}+\frac{73488136344821}{256}a^{8}-\frac{110190365143567}{64}a^{7}+\frac{31038754125565}{16}a^{6}+\frac{12847545887865}{4}a^{5}-13173625593680a^{4}+\frac{19\cdots 35}{1073741824}a^{3}-\frac{36\cdots 45}{268435456}a^{2}+\frac{33\cdots 89}{67108864}a-\frac{13\cdots 83}{16777216}$, $\frac{14\cdots 15}{536870912}a^{19}+\frac{66\cdots 31}{134217728}a^{18}+\frac{21\cdots 39}{33554432}a^{17}+\frac{49\cdots 71}{8388608}a^{16}+\frac{46\cdots 63}{2097152}a^{15}-\frac{28\cdots 65}{524288}a^{14}-\frac{21\cdots 85}{131072}a^{13}-\frac{91\cdots 29}{32768}a^{12}+\frac{24\cdots 29}{536870912}a^{11}+\frac{75\cdots 83}{134217728}a^{10}+\frac{10\cdots 65}{256}a^{9}-\frac{10\cdots 31}{64}a^{8}-\frac{73\cdots 95}{16}a^{7}-\frac{29\cdots 27}{4}a^{6}-90\!\cdots\!83a^{5}-74\!\cdots\!14a^{4}+\frac{64\cdots 33}{536870912}a^{3}-\frac{21\cdots 83}{134217728}a^{2}+\frac{27\cdots 47}{33554432}a-\frac{12\cdots 37}{8388608}$, $\frac{20\cdots 63}{1073741824}a^{19}+\frac{61\cdots 83}{268435456}a^{18}-\frac{56\cdots 09}{67108864}a^{17}+\frac{11\cdots 99}{16777216}a^{16}+\frac{53\cdots 95}{4194304}a^{15}-\frac{38\cdots 41}{1048576}a^{14}+\frac{56\cdots 95}{262144}a^{13}+\frac{43\cdots 71}{65536}a^{12}+\frac{18\cdots 33}{1073741824}a^{11}-\frac{18\cdots 57}{268435456}a^{10}-\frac{16\cdots 45}{1024}a^{9}+\frac{80\cdots 95}{256}a^{8}-\frac{31\cdots 57}{64}a^{7}-\frac{12\cdots 17}{16}a^{6}+\frac{51\cdots 79}{4}a^{5}+11\!\cdots\!10a^{4}-\frac{29\cdots 19}{1073741824}a^{3}+\frac{93\cdots 17}{268435456}a^{2}-\frac{11\cdots 05}{67108864}a+\frac{57\cdots 99}{16777216}$, $\frac{42\cdots 63}{536870912}a^{19}+\frac{15\cdots 71}{134217728}a^{18}+\frac{44\cdots 63}{33554432}a^{17}+\frac{91\cdots 75}{8388608}a^{16}+\frac{65\cdots 67}{2097152}a^{15}-\frac{50\cdots 89}{524288}a^{14}-\frac{33\cdots 17}{131072}a^{13}-\frac{13\cdots 25}{32768}a^{12}+\frac{70\cdots 09}{536870912}a^{11}-\frac{29\cdots 41}{134217728}a^{10}-\frac{84\cdots 89}{512}a^{9}-\frac{88\cdots 53}{128}a^{8}-\frac{40\cdots 81}{32}a^{7}-\frac{13\cdots 21}{8}a^{6}-\frac{327464306554419}{2}a^{5}-81048328843242a^{4}+\frac{18\cdots 49}{536870912}a^{3}-\frac{74\cdots 07}{134217728}a^{2}+\frac{10\cdots 15}{33554432}a-\frac{54\cdots 57}{8388608}$, $\frac{43\cdots 71}{536870912}a^{19}+\frac{89\cdots 63}{134217728}a^{18}+\frac{31\cdots 79}{33554432}a^{17}+\frac{15\cdots 83}{8388608}a^{16}+\frac{66\cdots 95}{2097152}a^{15}+\frac{24\cdots 75}{524288}a^{14}+\frac{74\cdots 43}{131072}a^{13}+\frac{17\cdots 31}{32768}a^{12}+\frac{77\cdots 73}{536870912}a^{11}-\frac{15\cdots 25}{134217728}a^{10}+\frac{59\cdots 03}{1024}a^{9}+\frac{26\cdots 87}{256}a^{8}+\frac{82\cdots 95}{64}a^{7}+\frac{18\cdots 75}{16}a^{6}+\frac{131709622229035}{4}a^{5}-127639436944660a^{4}+\frac{17\cdots 17}{536870912}a^{3}-\frac{12\cdots 91}{134217728}a^{2}+\frac{25\cdots 63}{33554432}a-\frac{17\cdots 89}{8388608}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 352917127553000000 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 352917127553000000 \cdot 4}{2\cdot\sqrt{4336370964135621995388739780608000000000000000000000}}\cr\approx \mathstrut & 1.02787002475893 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 + 175*x^12 - 280*x^11 + 112*x^10 + 4375*x^4 - 14000*x^3 + 16800*x^2 - 8960*x + 1792) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 + 175*x^12 - 280*x^11 + 112*x^10 + 4375*x^4 - 14000*x^3 + 16800*x^2 - 8960*x + 1792, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 175*x^12 - 280*x^11 + 112*x^10 + 4375*x^4 - 14000*x^3 + 16800*x^2 - 8960*x + 1792); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 175*x^12 - 280*x^11 + 112*x^10 + 4375*x^4 - 14000*x^3 + 16800*x^2 - 8960*x + 1792); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$A_5^4.D_4^2.C_2$ (as 20T1103):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 1658880000
The 665 conjugacy class representatives for $A_5^4.D_4^2.C_2$
Character table for $A_5^4.D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-14 +2 \sqrt{21}})\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 24 sibling: data not computed
Degree 40 siblings: data not computed
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 R R R ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.8.0.1}{8} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.2.4a2.2$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
2.2.8.48c6.377$x^{16} + 8 x^{15} + 48 x^{14} + 200 x^{13} + 628 x^{12} + 1528 x^{11} + 2992 x^{10} + 4800 x^{9} + 6413 x^{8} + 7168 x^{7} + 6728 x^{6} + 5264 x^{5} + 3412 x^{4} + 1784 x^{3} + 744 x^{2} + 240 x + 63$$8$$2$$48$16T1154$$[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, 4]^{4}$$
\(3\) Copy content Toggle raw display 3.1.4.3a1.2$x^{4} + 6$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.3.2.3a1.1$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
3.5.2.5a1.1$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 7 x + 1$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(5\) Copy content Toggle raw display 5.1.5.6a1.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$$[\frac{3}{2}]_{2}$$
5.1.5.5a1.2$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$$[\frac{5}{4}]_{4}$$
5.2.5.10a6.1$x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2280 x^{6} + 4064 x^{5} + 4560 x^{4} + 3220 x^{3} + 1450 x^{2} + 400 x + 57$$5$$2$$10$$(C_5^2 : C_4) : C_2$$$[\frac{5}{4}, \frac{5}{4}]_{4}^{2}$$
\(7\) Copy content Toggle raw display 7.1.20.19a1.1$x^{20} + 7$$20$$1$$19$20T18$$[\ ]_{20}^{4}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)