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Properties

Label 22.2.632...976.1
Degree $22$
Signature $[2, 10]$
Discriminant $6.321\times 10^{42}$
Root discriminant \(88.20\)
Ramified primes $2,7,11$
Class number $20$ (GRH)
Class group [20] (GRH)
Galois group $C_2^{10}.F_{11}$ (as 22T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 55*x^20 + 1177*x^18 + 12639*x^16 + 73898*x^14 + 237402*x^12 + 390258*x^10 + 246906*x^8 - 33011*x^6 - 40249*x^4 - 451*x^2 - 1)
 
gp: K = bnfinit(y^22 + 55*y^20 + 1177*y^18 + 12639*y^16 + 73898*y^14 + 237402*y^12 + 390258*y^10 + 246906*y^8 - 33011*y^6 - 40249*y^4 - 451*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 55*x^20 + 1177*x^18 + 12639*x^16 + 73898*x^14 + 237402*x^12 + 390258*x^10 + 246906*x^8 - 33011*x^6 - 40249*x^4 - 451*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 55*x^20 + 1177*x^18 + 12639*x^16 + 73898*x^14 + 237402*x^12 + 390258*x^10 + 246906*x^8 - 33011*x^6 - 40249*x^4 - 451*x^2 - 1)
 

\( x^{22} + 55 x^{20} + 1177 x^{18} + 12639 x^{16} + 73898 x^{14} + 237402 x^{12} + 390258 x^{10} + \cdots - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(6320614583435223575185822515817648777854976\) \(\medspace = 2^{38}\cdot 7^{10}\cdot 11^{22}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(88.20\)
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:  not computed
Ramified primes:   \(2\), \(7\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{44}a^{12}-\frac{3}{22}a^{10}-\frac{7}{44}a^{8}+\frac{1}{22}a^{6}-\frac{7}{44}a^{4}+\frac{4}{11}a^{2}+\frac{1}{44}$, $\frac{1}{44}a^{13}+\frac{5}{44}a^{11}-\frac{1}{4}a^{10}+\frac{1}{11}a^{9}-\frac{1}{4}a^{8}+\frac{1}{22}a^{7}-\frac{7}{44}a^{5}+\frac{5}{44}a^{3}+\frac{1}{4}a^{2}-\frac{5}{22}a+\frac{1}{4}$, $\frac{1}{44}a^{14}+\frac{1}{44}a^{10}+\frac{1}{11}a^{8}+\frac{5}{44}a^{6}+\frac{9}{22}a^{4}+\frac{9}{44}a^{2}+\frac{3}{22}$, $\frac{1}{44}a^{15}+\frac{1}{44}a^{11}+\frac{1}{11}a^{9}+\frac{5}{44}a^{7}+\frac{9}{22}a^{5}+\frac{9}{44}a^{3}+\frac{3}{22}a$, $\frac{1}{44}a^{16}+\frac{5}{22}a^{10}-\frac{5}{22}a^{8}-\frac{3}{22}a^{6}-\frac{3}{22}a^{4}+\frac{3}{11}a^{2}-\frac{1}{44}$, $\frac{1}{88}a^{17}-\frac{1}{88}a^{16}+\frac{5}{44}a^{11}-\frac{5}{44}a^{10}+\frac{3}{22}a^{9}-\frac{3}{22}a^{8}+\frac{2}{11}a^{7}-\frac{2}{11}a^{6}+\frac{2}{11}a^{5}-\frac{2}{11}a^{4}-\frac{5}{44}a^{3}+\frac{5}{44}a^{2}+\frac{43}{88}a-\frac{43}{88}$, $\frac{1}{88}a^{18}-\frac{1}{88}a^{16}+\frac{9}{44}a^{10}-\frac{7}{44}a^{8}-\frac{5}{22}a^{6}-\frac{1}{2}a^{4}+\frac{25}{88}a^{2}+\frac{35}{88}$, $\frac{1}{88}a^{19}-\frac{1}{88}a^{16}+\frac{3}{44}a^{11}+\frac{3}{22}a^{10}+\frac{5}{22}a^{9}+\frac{5}{44}a^{8}-\frac{1}{22}a^{7}-\frac{2}{11}a^{6}-\frac{7}{22}a^{5}-\frac{2}{11}a^{4}+\frac{37}{88}a^{3}-\frac{3}{22}a^{2}-\frac{4}{11}a+\frac{23}{88}$, $\frac{1}{10\!\cdots\!28}a^{20}+\frac{3013257342723}{10\!\cdots\!28}a^{18}+\frac{5489254673333}{506853095290564}a^{16}-\frac{3006629573045}{506853095290564}a^{14}+\frac{5209250624075}{506853095290564}a^{12}-\frac{28372247731547}{253426547645282}a^{10}-\frac{57162816319371}{253426547645282}a^{8}-\frac{16695858270279}{506853095290564}a^{6}-\frac{302880706394047}{10\!\cdots\!28}a^{4}-\frac{429433860712951}{10\!\cdots\!28}a^{2}-\frac{239307430839203}{506853095290564}$, $\frac{1}{10\!\cdots\!28}a^{21}+\frac{3013257342723}{10\!\cdots\!28}a^{19}-\frac{540879182665}{10\!\cdots\!28}a^{17}-\frac{1}{88}a^{16}-\frac{3006629573045}{506853095290564}a^{15}+\frac{5209250624075}{506853095290564}a^{13}+\frac{3092958928223}{126713273822641}a^{11}+\frac{3}{22}a^{10}-\frac{56728689992087}{506853095290564}a^{9}+\frac{5}{44}a^{8}-\frac{108850966504927}{506853095290564}a^{7}-\frac{2}{11}a^{6}-\frac{487190922863343}{10\!\cdots\!28}a^{5}-\frac{2}{11}a^{4}+\frac{446039667516205}{10\!\cdots\!28}a^{3}-\frac{3}{22}a^{2}-\frac{213668925503793}{10\!\cdots\!28}a+\frac{23}{88}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{20}$, which has order $20$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $a$, $\frac{10957034021973}{126713273822641}a^{21}+\frac{48\!\cdots\!17}{10\!\cdots\!28}a^{19}+\frac{10\!\cdots\!77}{10\!\cdots\!28}a^{17}+\frac{27\!\cdots\!83}{253426547645282}a^{15}+\frac{32\!\cdots\!77}{506853095290564}a^{13}+\frac{10\!\cdots\!85}{506853095290564}a^{11}+\frac{42\!\cdots\!18}{126713273822641}a^{9}+\frac{26\!\cdots\!74}{126713273822641}a^{7}-\frac{15\!\cdots\!75}{506853095290564}a^{5}-\frac{31\!\cdots\!73}{92155108234648}a^{3}-\frac{64\!\cdots\!25}{10\!\cdots\!28}a$, $\frac{2572978654690}{126713273822641}a^{20}+\frac{283023529729063}{253426547645282}a^{18}+\frac{60\!\cdots\!95}{253426547645282}a^{16}+\frac{13\!\cdots\!31}{506853095290564}a^{14}+\frac{76\!\cdots\!43}{506853095290564}a^{12}+\frac{24\!\cdots\!27}{506853095290564}a^{10}+\frac{40\!\cdots\!15}{506853095290564}a^{8}+\frac{25\!\cdots\!33}{506853095290564}a^{6}-\frac{35\!\cdots\!35}{506853095290564}a^{4}-\frac{41\!\cdots\!33}{506853095290564}a^{2}-\frac{24\!\cdots\!21}{506853095290564}$, $\frac{51432792478343}{506853095290564}a^{21}+\frac{514261074560519}{92155108234648}a^{19}+\frac{12\!\cdots\!61}{10\!\cdots\!28}a^{17}+\frac{32\!\cdots\!43}{253426547645282}a^{15}+\frac{18\!\cdots\!57}{253426547645282}a^{13}+\frac{12\!\cdots\!43}{506853095290564}a^{11}+\frac{19\!\cdots\!17}{506853095290564}a^{9}+\frac{62\!\cdots\!75}{253426547645282}a^{7}-\frac{17\!\cdots\!73}{506853095290564}a^{5}-\frac{37\!\cdots\!97}{92155108234648}a^{3}-\frac{21\!\cdots\!15}{10\!\cdots\!28}a$, $\frac{176926430301759}{10\!\cdots\!28}a^{21}+\frac{97\!\cdots\!17}{10\!\cdots\!28}a^{19}+\frac{26\!\cdots\!96}{126713273822641}a^{17}+\frac{55\!\cdots\!19}{253426547645282}a^{15}+\frac{65\!\cdots\!05}{506853095290564}a^{13}+\frac{20\!\cdots\!43}{506853095290564}a^{11}+\frac{85\!\cdots\!54}{126713273822641}a^{9}+\frac{53\!\cdots\!88}{126713273822641}a^{7}-\frac{64\!\cdots\!53}{10\!\cdots\!28}a^{5}-\frac{70\!\cdots\!47}{10\!\cdots\!28}a^{3}+\frac{55\!\cdots\!39}{253426547645282}a$, $\frac{13226664538913}{506853095290564}a^{20}+\frac{181861256419006}{126713273822641}a^{18}+\frac{38\!\cdots\!35}{126713273822641}a^{16}+\frac{83\!\cdots\!65}{253426547645282}a^{14}+\frac{97\!\cdots\!99}{506853095290564}a^{12}+\frac{15\!\cdots\!61}{253426547645282}a^{10}+\frac{46\!\cdots\!71}{46077554117324}a^{8}+\frac{81\!\cdots\!56}{126713273822641}a^{6}-\frac{22\!\cdots\!85}{253426547645282}a^{4}-\frac{12\!\cdots\!90}{11519388529331}a^{2}-\frac{39\!\cdots\!79}{506853095290564}$, $\frac{145506972600265}{506853095290564}a^{21}+\frac{16\!\cdots\!89}{10\!\cdots\!28}a^{19}+\frac{34\!\cdots\!67}{10\!\cdots\!28}a^{17}+\frac{18\!\cdots\!41}{506853095290564}a^{15}+\frac{10\!\cdots\!97}{506853095290564}a^{13}+\frac{86\!\cdots\!39}{126713273822641}a^{11}+\frac{28\!\cdots\!91}{253426547645282}a^{9}+\frac{35\!\cdots\!45}{506853095290564}a^{7}-\frac{12\!\cdots\!76}{126713273822641}a^{5}-\frac{11\!\cdots\!53}{10\!\cdots\!28}a^{3}-\frac{78\!\cdots\!75}{10\!\cdots\!28}a$, $\frac{89198541476139}{92155108234648}a^{21}+\frac{49\!\cdots\!99}{92155108234648}a^{19}+\frac{28\!\cdots\!71}{253426547645282}a^{17}+\frac{30\!\cdots\!45}{253426547645282}a^{15}+\frac{36\!\cdots\!15}{506853095290564}a^{13}+\frac{11\!\cdots\!81}{506853095290564}a^{11}+\frac{95\!\cdots\!25}{253426547645282}a^{9}+\frac{27\!\cdots\!56}{11519388529331}a^{7}-\frac{33\!\cdots\!35}{10\!\cdots\!28}a^{5}-\frac{35\!\cdots\!85}{92155108234648}a^{3}-\frac{80\!\cdots\!79}{253426547645282}a$, $\frac{23011077324471}{10\!\cdots\!28}a^{21}+\frac{158275921054231}{126713273822641}a^{19}+\frac{27\!\cdots\!67}{10\!\cdots\!28}a^{17}+\frac{36\!\cdots\!03}{126713273822641}a^{15}+\frac{21\!\cdots\!84}{126713273822641}a^{13}+\frac{13\!\cdots\!41}{253426547645282}a^{11}+\frac{22\!\cdots\!85}{253426547645282}a^{9}+\frac{14\!\cdots\!21}{253426547645282}a^{7}-\frac{71\!\cdots\!55}{10\!\cdots\!28}a^{5}-\frac{11\!\cdots\!77}{126713273822641}a^{3}-\frac{17\!\cdots\!75}{10\!\cdots\!28}a$, $\frac{827647207008109}{10\!\cdots\!28}a^{21}+\frac{11\!\cdots\!95}{253426547645282}a^{19}+\frac{97\!\cdots\!69}{10\!\cdots\!28}a^{17}+\frac{52\!\cdots\!75}{506853095290564}a^{15}+\frac{76\!\cdots\!74}{126713273822641}a^{13}+\frac{97\!\cdots\!17}{506853095290564}a^{11}+\frac{40\!\cdots\!04}{126713273822641}a^{9}+\frac{91\!\cdots\!39}{46077554117324}a^{7}-\frac{28\!\cdots\!57}{10\!\cdots\!28}a^{5}-\frac{16\!\cdots\!75}{506853095290564}a^{3}-\frac{96\!\cdots\!61}{10\!\cdots\!28}a$, $\frac{66\!\cdots\!71}{253426547645282}a^{21}+\frac{15\!\cdots\!77}{10\!\cdots\!28}a^{20}+\frac{14\!\cdots\!47}{10\!\cdots\!28}a^{19}+\frac{43\!\cdots\!27}{506853095290564}a^{18}+\frac{15\!\cdots\!31}{506853095290564}a^{17}+\frac{23\!\cdots\!35}{126713273822641}a^{16}+\frac{39\!\cdots\!70}{11519388529331}a^{15}+\frac{10\!\cdots\!29}{506853095290564}a^{14}+\frac{25\!\cdots\!75}{126713273822641}a^{13}+\frac{61\!\cdots\!77}{506853095290564}a^{12}+\frac{17\!\cdots\!37}{253426547645282}a^{11}+\frac{10\!\cdots\!73}{253426547645282}a^{10}+\frac{64\!\cdots\!91}{506853095290564}a^{9}+\frac{38\!\cdots\!09}{506853095290564}a^{8}+\frac{13\!\cdots\!37}{126713273822641}a^{7}+\frac{30\!\cdots\!29}{46077554117324}a^{6}+\frac{70\!\cdots\!77}{23038777058662}a^{5}+\frac{18\!\cdots\!05}{10\!\cdots\!28}a^{4}+\frac{33\!\cdots\!89}{10\!\cdots\!28}a^{3}+\frac{10\!\cdots\!23}{506853095290564}a^{2}+\frac{18\!\cdots\!07}{253426547645282}a+\frac{20\!\cdots\!29}{46077554117324}$ Copy content Toggle raw display (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:  \( 1583985040230 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 1583985040230 \cdot 20}{2\cdot\sqrt{6320614583435223575185822515817648777854976}}\cr\approx \mathstrut & 2.41674092300409 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + 55*x^20 + 1177*x^18 + 12639*x^16 + 73898*x^14 + 237402*x^12 + 390258*x^10 + 246906*x^8 - 33011*x^6 - 40249*x^4 - 451*x^2 - 1)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^22 + 55*x^20 + 1177*x^18 + 12639*x^16 + 73898*x^14 + 237402*x^12 + 390258*x^10 + 246906*x^8 - 33011*x^6 - 40249*x^4 - 451*x^2 - 1, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 + 55*x^20 + 1177*x^18 + 12639*x^16 + 73898*x^14 + 237402*x^12 + 390258*x^10 + 246906*x^8 - 33011*x^6 - 40249*x^4 - 451*x^2 - 1);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 55*x^20 + 1177*x^18 + 12639*x^16 + 73898*x^14 + 237402*x^12 + 390258*x^10 + 246906*x^8 - 33011*x^6 - 40249*x^4 - 451*x^2 - 1);
 
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

$C_2^{10}.F_{11}$ (as 22T34):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 112640
The 44 conjugacy class representatives for $C_2^{10}.F_{11}$
Character table for $C_2^{10}.F_{11}$

Intermediate fields

11.11.4910318845910094848.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 22 sibling: data not computed
Degree 44 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 ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ R R ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 Deg $22$$22$$1$$38$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.20.10.1$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(11\) Copy content Toggle raw display 11.11.11.6$x^{11} + 11 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$
11.11.11.6$x^{11} + 11 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$