Properties

Label 82110.x1
Conductor 8211082110
Discriminant 1.605×10131.605\times 10^{13}
j-invariant 92358074039307904916049455024050 \frac{923580740393079049}{16049455024050}
CM no
Rank 11
Torsion structure Z/2Z\Z/{2}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x320289x+1093786y^2+xy+y=x^3-20289x+1093786 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x320289xz2+1093786z3y^2z+xyz+yz^2=x^3-20289xz^2+1093786z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x326293923x+51110573022y^2=x^3-26293923x+51110573022 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 1, -20289, 1093786])
 
gp: E = ellinit([1, 0, 1, -20289, 1093786])
 
magma: E := EllipticCurve([1, 0, 1, -20289, 1093786]);
 
oscar: E = elliptic_curve([1, 0, 1, -20289, 1093786])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

ZZ/2Z\Z \oplus \Z/{2}\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(68,138)(68, 138)0.265844682837809808810286663670.26584468283780980881028666367\infty
(295/4,299/8)(295/4, -299/8)0022

Integral points

(44,513) \left(44, 513\right) , (44,558) \left(44, -558\right) , (68,138) \left(68, 138\right) , (68,207) \left(68, -207\right) , (94,92) \left(94, 92\right) , (94,187) \left(94, -187\right) , (206,2277) \left(206, 2277\right) , (206,2484) \left(206, -2484\right) , (758,20148) \left(758, 20148\right) , (758,20907) \left(758, -20907\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  =  1604945502405016049455024050 = 23452721722342 \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \cdot 17^{2} \cdot 23^{4}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  92358074039307904916049455024050 \frac{923580740393079049}{16049455024050}  = 213452721722342933358132^{-1} \cdot 3^{-4} \cdot 5^{-2} \cdot 7^{-2} \cdot 17^{-2} \cdot 23^{-4} \cdot 29^{3} \cdot 33581^{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}} ≈ 1.33071524651917208996630379171.3307152465191720899663037917
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.33071524651917208996630379171.3307152465191720899663037917
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.07878481302004661.0787848130200466
Szpiro ratio: σm\sigma_{m} ≈ 3.6556831556397113.655683155639711

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 0.265844682837809808810286663670.26584468283780980881028666367
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.697702976500175138745654851920.69770297650017513874565485192
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 = 128 128  = 12222222 1\cdot2^{2}\cdot2\cdot2\cdot2\cdot2^{2}
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}} = 22
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) ≈ 5.93538004808591776166797296035.9353800480859177616679729603
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    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

5.935380048L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.6977030.265845128225.935380048\displaystyle 5.935380048 \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 0.697703 \cdot 0.265845 \cdot 128}{2^2} \approx 5.935380048

# 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.x

qq2+q3+q4q5q6+q7q8+q9+q102q11+q12+4q13q14q15+q16+q17q18+2q19+O(q20) q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - 2 q^{11} + q^{12} + 4 q^{13} - q^{14} - q^{15} + q^{16} + q^{17} - q^{18} + 2 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: 270336
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 11 I1I_{1} nonsplit multiplicative 1 1 1 1
33 44 I4I_{4} split multiplicative -1 1 4 4
55 22 I2I_{2} nonsplit multiplicative 1 1 2 2
77 22 I2I_{2} split multiplicative -1 1 2 2
1717 22 I2I_{2} split multiplicative -1 1 2 2
2323 44 I4I_{4} split multiplicative -1 1 4 4

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 8.6.0.6

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 = [[409, 4, 818, 9], [1, 2, 2, 5], [121, 834, 832, 119], [1, 4, 0, 1], [1, 0, 4, 1], [3, 4, 8, 11], [949, 4, 948, 5], [785, 4, 618, 9], [2, 1, 475, 0]]
 
GL(2,Integers(952)).subgroup(gens)
 
Gens := [[409, 4, 818, 9], [1, 2, 2, 5], [121, 834, 832, 119], [1, 4, 0, 1], [1, 0, 4, 1], [3, 4, 8, 11], [949, 4, 948, 5], [785, 4, 618, 9], [2, 1, 475, 0]];
 
sub<GL(2,Integers(952))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 952=23717 952 = 2^{3} \cdot 7 \cdot 17 , index 1212, genus 00, and generators

(40948189),(1225),(121834832119),(1401),(1041),(34811),(94949485),(78546189),(214750)\left(\begin{array}{rr} 409 & 4 \\ 818 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 121 & 834 \\ 832 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 949 & 4 \\ 948 & 5 \end{array}\right),\left(\begin{array}{rr} 785 & 4 \\ 618 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 475 & 0 \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[952])K:=\Q(E[952]) is a degree-2021444812820214448128 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/952Z)\GL_2(\Z/952\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 1 1
33 split multiplicative 44 27370=2571723 27370 = 2 \cdot 5 \cdot 7 \cdot 17 \cdot 23
55 nonsplit 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 split 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.
Its isogeny class 82110.x consists of 2 curves linked by isogenies of degree 2.

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/2Z\cong \Z/{2}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(2)\Q(\sqrt{2}) Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
44 4.0.453152.2 Z/4Z\Z/4\Z not in database
88 deg 8 Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
88 8.0.13142191046656.12 Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
88 deg 8 Z/6Z\Z/6\Z not in database
1616 deg 16 Z/8Z\Z/8\Z not in database
1616 deg 16 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\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 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit split nonsplit split ord ord split ord split ss ss ord ord ord ord
λ\lambda-invariant(s) 8 4 1 2 1 1 2 1 2 1,1 1,1 3 1 1 1
μ\mu-invariant(s) 0 0 0 0 0 0 0 0 0 0,0 0,0 0 0 0 0

pp-adic regulators

pp-adic regulators are not yet computed for curves that are not Γ0\Gamma_0-optimal.