Properties

Label 82110.o3
Conductor 8211082110
Discriminant 5.316×1018-5.316\times 10^{18}
j-invariant 21095829373515554725215315560665303840000 -\frac{2109582937351555472521}{5315560665303840000}
CM no
Rank 00
Torsion structure Z/4Z\Z/{4}\Z

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy=x3+x2267192x123117504y^2+xy=x^3+x^2-267192x-123117504 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz=x3+x2z267192xz2123117504z3y^2z+xyz=x^3+x^2z-267192xz^2-123117504z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3346281507x5738976047394y^2=x^3-346281507x-5738976047394 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 1, 0, -267192, -123117504])
 
gp: E = ellinit([1, 1, 0, -267192, -123117504])
 
magma: E := EllipticCurve([1, 1, 0, -267192, -123117504]);
 
oscar: E = elliptic_curve([1, 1, 0, -267192, -123117504])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z/4Z\Z/{4}\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(1712,65784)(1712, 65784)0044

Integral points

(1712,65784) \left(1712, 65784\right) , (1712,67496) \left(1712, -67496\right) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  82110 82110  = 235717232 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  5315560665303840000-5315560665303840000 = 1283547817423-1 \cdot 2^{8} \cdot 3 \cdot 5^{4} \cdot 7^{8} \cdot 17^{4} \cdot 23
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  21095829373515554725215315560665303840000 -\frac{2109582937351555472521}{5315560665303840000}  = 12831547811313617423168993-1 \cdot 2^{-8} \cdot 3^{-1} \cdot 5^{-4} \cdot 7^{-8} \cdot 11^{3} \cdot 13^{6} \cdot 17^{-4} \cdot 23^{-1} \cdot 6899^{3}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 2.28093534075576653730043551082.2809353407557665373004355108
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 2.28093534075576653730043551082.2809353407557665373004355108
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.00537600359278721.0053760035927872
Szpiro ratio: σm\sigma_{m} ≈ 4.4874041633442514.487404163344251

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.0977651872028098120228635879800.097765187202809812022863587980
comment: Real Period
 
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 256 256  = 212223221 2\cdot1\cdot2^{2}\cdot2^{3}\cdot2^{2}\cdot1
comment: Tamagawa numbers
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
oscar: tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 44
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 1.56424299524495699236581740771.5642429952449569923658174077
comment: Special L-value
 
r = E.rank();
 
E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  =  11    (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

1.564242995L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.0977651.000000256421.564242995\displaystyle 1.564242995 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.097765 \cdot 1.000000 \cdot 256}{4^2} \approx 1.564242995

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
 
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
 
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
 
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
 
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
 
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
 
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
 
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form 82110.2.a.o

qq2q3+q4+q5+q6+q7q8+q9q10q122q13q14q15+q16+q17q18+4q19+O(q20) q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} - q^{12} - 2 q^{13} - q^{14} - q^{15} + q^{16} + q^{17} - q^{18} + 4 q^{19} + O(q^{20}) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ actual modular form, use for small N
 
[mf,F] = mffromell(E)
 
Ser(mfcoefs(mf,20),q)
 
\\ or just the series
 
Ser(ellan(E,20),q)*q
 
magma: ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 2031616
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: no
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is semistable. There are 6 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
22 22 I8I_{8} nonsplit multiplicative 1 1 8 8
33 11 I1I_{1} nonsplit multiplicative 1 1 1 1
55 44 I4I_{4} split multiplicative -1 1 4 4
77 88 I8I_{8} split multiplicative -1 1 8 8
1717 44 I4I_{4} split multiplicative -1 1 4 4
2323 11 I1I_{1} nonsplit multiplicative 1 1 1 1

comment: Local data
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
22 2B 4.12.0.7

comment: mod p Galois image
 
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

gens = [[1, 0, 8, 1], [29328, 5873, 29353, 29400], [1, 8, 0, 1], [1, 4, 4, 17], [41059, 41058, 29338, 5875], [2044, 1, 23, 6], [16561, 8, 19324, 33], [37537, 8, 9388, 33], [7, 6, 46914, 46915], [15644, 1, 31303, 6], [46913, 8, 46912, 9]]
 
GL(2,Integers(46920)).subgroup(gens)
 
Gens := [[1, 0, 8, 1], [29328, 5873, 29353, 29400], [1, 8, 0, 1], [1, 4, 4, 17], [41059, 41058, 29338, 5875], [2044, 1, 23, 6], [16561, 8, 19324, 33], [37537, 8, 9388, 33], [7, 6, 46914, 46915], [15644, 1, 31303, 6], [46913, 8, 46912, 9]];
 
sub<GL(2,Integers(46920))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 46920=23351723 46920 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \cdot 23 , index 4848, genus 00, and generators

(1081),(2932858732935329400),(1801),(14417),(4105941058293385875),(20441236),(1656181932433),(375378938833),(764691446915),(156441313036),(469138469129)\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 29328 & 5873 \\ 29353 & 29400 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 41059 & 41058 \\ 29338 & 5875 \end{array}\right),\left(\begin{array}{rr} 2044 & 1 \\ 23 & 6 \end{array}\right),\left(\begin{array}{rr} 16561 & 8 \\ 19324 & 33 \end{array}\right),\left(\begin{array}{rr} 37537 & 8 \\ 9388 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 46914 & 46915 \end{array}\right),\left(\begin{array}{rr} 15644 & 1 \\ 31303 & 6 \end{array}\right),\left(\begin{array}{rr} 46913 & 8 \\ 46912 & 9 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[46920])K:=\Q(E[46920]) is a degree-1543043907846144015430439078461440 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/46920Z)\GL_2(\Z/46920\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 nonsplit multiplicative 44 69=323 69 = 3 \cdot 23
33 nonsplit multiplicative 44 27370=2571723 27370 = 2 \cdot 5 \cdot 7 \cdot 17 \cdot 23
55 split multiplicative 66 16422=2371723 16422 = 2 \cdot 3 \cdot 7 \cdot 17 \cdot 23
77 split multiplicative 88 11730=2351723 11730 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 23
1717 split multiplicative 1818 4830=235723 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23
2323 nonsplit multiplicative 2424 3570=235717 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2 and 4.
Its isogeny class 82110.o consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

This elliptic curve is its own minimal quadratic twist.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} Z/4Z\cong \Z/{4}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(69)\Q(\sqrt{-69}) Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
44 4.2.7976400.8 Z/8Z\Z/8\Z not in database
88 8.0.7072524735676416.8 Z/4ZZ/4Z\Z/4\Z \oplus \Z/4\Z not in database
88 deg 8 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 deg 8 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 deg 8 Z/12Z\Z/12\Z not in database
1616 deg 16 Z/16Z\Z/16\Z not in database
1616 deg 16 Z/2ZZ/12Z\Z/2\Z \oplus \Z/12\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

pp 2 3 5 7 17 23
Reduction type nonsplit nonsplit split split split nonsplit
λ\lambda-invariant(s) 8 0 1 1 1 0
μ\mu-invariant(s) 2 0 0 0 0 0

All Iwasawa λ\lambda and μ\mu-invariants for primes p3p\ge 3 of good reduction are zero.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.