y 2 + x y = x 3 + x 2 − 2326184142 x − 43184119407564 y^2+xy=x^3+x^2-2326184142x-43184119407564 y 2 + x y = x 3 + x 2 − 2 3 2 6 1 8 4 1 4 2 x − 4 3 1 8 4 1 1 9 4 0 7 5 6 4
(homogenize , simplify )
y 2 z + x y z = x 3 + x 2 z − 2326184142 x z 2 − 43184119407564 z 3 y^2z+xyz=x^3+x^2z-2326184142xz^2-43184119407564z^3 y 2 z + x y z = x 3 + x 2 z − 2 3 2 6 1 8 4 1 4 2 x z 2 − 4 3 1 8 4 1 1 9 4 0 7 5 6 4 z 3
(dehomogenize , simplify )
y 2 = x 3 − 3014734648707 x − 2014753054059578754 y^2=x^3-3014734648707x-2014753054059578754 y 2 = x 3 − 3 0 1 4 7 3 4 6 4 8 7 0 7 x − 2 0 1 4 7 5 3 0 5 4 0 5 9 5 7 8 7 5 4
(homogenize , minimize )
sage: E = EllipticCurve([1, 1, 0, -2326184142, -43184119407564])
gp: E = ellinit([1, 1, 0, -2326184142, -43184119407564])
magma: E := EllipticCurve([1, 1, 0, -2326184142, -43184119407564]);
oscar: E = elliptic_curve([1, 1, 0, -2326184142, -43184119407564])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Z \Z Z
magma: MordellWeilGroup(E);
P P P h ^ ( P ) \hat{h}(P) h ^ ( P ) Order
( 6996409804533344795784282062906976115685293541090118077792677481521744898824956341948420389027529243362355394144437308365756742128503143382509094070546902972840790401877668975 / 125497966090630596192190099740585100188948813213759943286340035734525663240912348233389602276999588311096369538734689531536603611246430342981635668205316837247825604104881 , 854203150932830263075905593409543125484607764898592251883306775348081740339765606281291117062203946576785576151884920022128714274686908936317582762572740194005992339476217699135894589116129158627297170191381920684339163227338739732164320819266006641627307678999 / 1405901942816196429865092410607312454628190087502355293557351544334208304608342112959247303167258874715446630895292819695383314562648186104975987863918462010779633673778709442051114949515013615715805360838607658354514701340707810140876978790353092918375529 ) (6996409804533344795784282062906976115685293541090118077792677481521744898824956341948420389027529243362355394144437308365756742128503143382509094070546902972840790401877668975/125497966090630596192190099740585100188948813213759943286340035734525663240912348233389602276999588311096369538734689531536603611246430342981635668205316837247825604104881, 854203150932830263075905593409543125484607764898592251883306775348081740339765606281291117062203946576785576151884920022128714274686908936317582762572740194005992339476217699135894589116129158627297170191381920684339163227338739732164320819266006641627307678999/1405901942816196429865092410607312454628190087502355293557351544334208304608342112959247303167258874715446630895292819695383314562648186104975987863918462010779633673778709442051114949515013615715805360838607658354514701340707810140876978790353092918375529) ( 6 9 9 6 4 0 9 8 0 4 5 3 3 3 4 4 7 9 5 7 8 4 2 8 2 0 6 2 9 0 6 9 7 6 1 1 5 6 8 5 2 9 3 5 4 1 0 9 0 1 1 8 0 7 7 7 9 2 6 7 7 4 8 1 5 2 1 7 4 4 8 9 8 8 2 4 9 5 6 3 4 1 9 4 8 4 2 0 3 8 9 0 2 7 5 2 9 2 4 3 3 6 2 3 5 5 3 9 4 1 4 4 4 3 7 3 0 8 3 6 5 7 5 6 7 4 2 1 2 8 5 0 3 1 4 3 3 8 2 5 0 9 0 9 4 0 7 0 5 4 6 9 0 2 9 7 2 8 4 0 7 9 0 4 0 1 8 7 7 6 6 8 9 7 5 / 1 2 5 4 9 7 9 6 6 0 9 0 6 3 0 5 9 6 1 9 2 1 9 0 0 9 9 7 4 0 5 8 5 1 0 0 1 8 8 9 4 8 8 1 3 2 1 3 7 5 9 9 4 3 2 8 6 3 4 0 0 3 5 7 3 4 5 2 5 6 6 3 2 4 0 9 1 2 3 4 8 2 3 3 3 8 9 6 0 2 2 7 6 9 9 9 5 8 8 3 1 1 0 9 6 3 6 9 5 3 8 7 3 4 6 8 9 5 3 1 5 3 6 6 0 3 6 1 1 2 4 6 4 3 0 3 4 2 9 8 1 6 3 5 6 6 8 2 0 5 3 1 6 8 3 7 2 4 7 8 2 5 6 0 4 1 0 4 8 8 1 , 8 5 4 2 0 3 1 5 0 9 3 2 8 3 0 2 6 3 0 7 5 9 0 5 5 9 3 4 0 9 5 4 3 1 2 5 4 8 4 6 0 7 7 6 4 8 9 8 5 9 2 2 5 1 8 8 3 3 0 6 7 7 5 3 4 8 0 8 1 7 4 0 3 3 9 7 6 5 6 0 6 2 8 1 2 9 1 1 1 7 0 6 2 2 0 3 9 4 6 5 7 6 7 8 5 5 7 6 1 5 1 8 8 4 9 2 0 0 2 2 1 2 8 7 1 4 2 7 4 6 8 6 9 0 8 9 3 6 3 1 7 5 8 2 7 6 2 5 7 2 7 4 0 1 9 4 0 0 5 9 9 2 3 3 9 4 7 6 2 1 7 6 9 9 1 3 5 8 9 4 5 8 9 1 1 6 1 2 9 1 5 8 6 2 7 2 9 7 1 7 0 1 9 1 3 8 1 9 2 0 6 8 4 3 3 9 1 6 3 2 2 7 3 3 8 7 3 9 7 3 2 1 6 4 3 2 0 8 1 9 2 6 6 0 0 6 6 4 1 6 2 7 3 0 7 6 7 8 9 9 9 / 1 4 0 5 9 0 1 9 4 2 8 1 6 1 9 6 4 2 9 8 6 5 0 9 2 4 1 0 6 0 7 3 1 2 4 5 4 6 2 8 1 9 0 0 8 7 5 0 2 3 5 5 2 9 3 5 5 7 3 5 1 5 4 4 3 3 4 2 0 8 3 0 4 6 0 8 3 4 2 1 1 2 9 5 9 2 4 7 3 0 3 1 6 7 2 5 8 8 7 4 7 1 5 4 4 6 6 3 0 8 9 5 2 9 2 8 1 9 6 9 5 3 8 3 3 1 4 5 6 2 6 4 8 1 8 6 1 0 4 9 7 5 9 8 7 8 6 3 9 1 8 4 6 2 0 1 0 7 7 9 6 3 3 6 7 3 7 7 8 7 0 9 4 4 2 0 5 1 1 1 4 9 4 9 5 1 5 0 1 3 6 1 5 7 1 5 8 0 5 3 6 0 8 3 8 6 0 7 6 5 8 3 5 4 5 1 4 7 0 1 3 4 0 7 0 7 8 1 0 1 4 0 8 7 6 9 7 8 7 9 0 3 5 3 0 9 2 9 1 8 3 7 5 5 2 9 ) 403.00033034405929621846242365 403.00033034405929621846242365 4 0 3 . 0 0 0 3 3 0 3 4 4 0 5 9 2 9 6 2 1 8 4 6 2 4 2 3 6 5 ∞ \infty ∞
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor :
N N N
=
111090 111090 1 1 1 0 9 0 = 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 2 3 2 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2} 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 2 3 2
sage: E.conductor().factor()
Discriminant :
Δ \Delta Δ
=
− 262014822331562022965760 -262014822331562022965760 − 2 6 2 0 1 4 8 2 2 3 3 1 5 6 2 0 2 2 9 6 5 7 6 0 = − 1 ⋅ 2 9 ⋅ 3 ⋅ 5 ⋅ 7 7 ⋅ 2 3 10 -1 \cdot 2^{9} \cdot 3 \cdot 5 \cdot 7^{7} \cdot 23^{10} − 1 ⋅ 2 9 ⋅ 3 ⋅ 5 ⋅ 7 7 ⋅ 2 3 1 0
sage: E.discriminant().factor()
j-invariant :
j j j
=
− 33602966923620213529 6324810240 -\frac{33602966923620213529}{6324810240} − 6 3 2 4 8 1 0 2 4 0 3 3 6 0 2 9 6 6 9 2 3 6 2 0 2 1 3 5 2 9 = − 1 ⋅ 2 − 9 ⋅ 3 − 1 ⋅ 5 − 1 ⋅ 7 − 7 ⋅ 2 3 2 ⋅ 3 1 3 ⋅ 6 1 3 ⋅ 21 1 3 -1 \cdot 2^{-9} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-7} \cdot 23^{2} \cdot 31^{3} \cdot 61^{3} \cdot 211^{3} − 1 ⋅ 2 − 9 ⋅ 3 − 1 ⋅ 5 − 1 ⋅ 7 − 7 ⋅ 2 3 2 ⋅ 3 1 3 ⋅ 6 1 3 ⋅ 2 1 1 3
sage: E.j_invariant().factor()
Endomorphism ring :
E n d ( E ) \mathrm{End}(E) E n d ( E ) = Z \Z Z
Geometric endomorphism ring :
E n d ( E Q ‾ ) \mathrm{End}(E_{\overline{\Q}}) E n d ( E Q )
=
Z \Z Z
(no potential complex multiplication )
magma: HasComplexMultiplication(E);
Sato-Tate group :
S T ( E ) \mathrm{ST}(E) S T ( E ) = S U ( 2 ) \mathrm{SU}(2) S U ( 2 )
Faltings height :
h F a l t i n g s h_{\mathrm{Faltings}} h F a l t i n g s ≈ 3.8863523513631375804957733745 3.8863523513631375804957733745 3 . 8 8 6 3 5 2 3 5 1 3 6 3 1 3 7 5 8 0 4 9 5 7 7 3 3 7 4 5
magma: FaltingsHeight(E);
oscar: faltings_height(E)
Stable Faltings height :
h s t a b l e h_{\mathrm{stable}} h s t a b l e ≈ 1.2734405047555128381568126813 1.2734405047555128381568126813 1 . 2 7 3 4 4 0 5 0 4 7 5 5 5 1 2 8 3 8 1 5 6 8 1 2 6 8 1 3
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
a b c abc a b c quality :
Q Q Q ≈ 1.025108410651787 1.025108410651787 1 . 0 2 5 1 0 8 4 1 0 6 5 1 7 8 7
Szpiro ratio :
σ m \sigma_{m} σ m ≈ 6.568725957216232 6.568725957216232 6 . 5 6 8 7 2 5 9 5 7 2 1 6 2 3 2
Analytic rank :
r a n r_{\mathrm{an}} r a n = 1 1 1
Mordell-Weil rank :
r r r = 1 1 1
gp: [lower,upper] = ellrank(E)
Regulator :
R e g ( E / Q ) \mathrm{Reg}(E/\Q) R e g ( E / Q ) ≈ 403.00033034405929621846242365 403.00033034405929621846242365 4 0 3 . 0 0 0 3 3 0 3 4 4 0 5 9 2 9 6 2 1 8 4 6 2 4 2 3 6 5
G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
Real period :
Ω \Omega Ω ≈ 0.010869474084102693366379149467 0.010869474084102693366379149467 0 . 0 1 0 8 6 9 4 7 4 0 8 4 1 0 2 6 9 3 3 6 6 3 7 9 1 4 9 4 6 7
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 :
∏ p c p \prod_{p}c_p ∏ p c p = 1 1 1
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 ) t o r \#E(\Q)_{\mathrm{tor}} # E ( Q ) t o r = 1 1 1
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
Special value :
L ′ ( E , 1 ) L'(E,1) L ′ ( E , 1 ) ≈ 4.3804016465595767841807972639 4.3804016465595767841807972639 4 . 3 8 0 4 0 1 6 4 6 5 5 9 5 7 6 7 8 4 1 8 0 7 9 7 2 6 3 9
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 :
Шa n {}_{\mathrm{an}} a n
≈
1 1 1
(rounded )
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
4.380401647 ≈ L ′ ( E , 1 ) = # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p # E ( Q ) t o r 2 ≈ 1 ⋅ 0.010869 ⋅ 403.000330 ⋅ 1 1 2 ≈ 4.380401647 \displaystyle 4.380401647 \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 0.010869 \cdot 403.000330 \cdot 1}{1^2} \approx 4.380401647 4 . 3 8 0 4 0 1 6 4 7 ≈ L ′ ( E , 1 ) = # E ( Q ) t o r 2 # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p ≈ 1 2 1 ⋅ 0 . 0 1 0 8 6 9 ⋅ 4 0 3 . 0 0 0 3 3 0 ⋅ 1 ≈ 4 . 3 8 0 4 0 1 6 4 7
# 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 form
111090.2.a.l
q − q 2 − q 3 + q 4 + q 5 + q 6 − q 7 − q 8 + q 9 − q 10 + 4 q 11 − q 12 − 2 q 13 + q 14 − q 15 + q 16 − q 18 − 2 q 19 + O ( q 20 ) q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} - q^{18} - 2 q^{19} + O(q^{20}) q − q 2 − q 3 + q 4 + q 5 + q 6 − q 7 − q 8 + q 9 − q 1 0 + 4 q 1 1 − q 1 2 − 2 q 1 3 + q 1 4 − q 1 5 + q 1 6 − q 1 8 − 2 q 1 9 + O ( q 2 0 )
\\ 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
For more coefficients, see the Downloads section to the right.
This elliptic curve is not semistable .
There
are 5 primes p p p
of bad reduction :
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)]
The ℓ \ell ℓ -adic Galois representation has maximal image
for all primes ℓ \ell ℓ .
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
gens = [[631, 2, 631, 3], [839, 2, 838, 3], [1, 0, 2, 1], [1, 2, 0, 1], [337, 2, 337, 3], [281, 2, 281, 3], [241, 2, 241, 3], [421, 2, 421, 3], [1, 1, 839, 0]]
GL(2,Integers(840)).subgroup(gens)
Gens := [[631, 2, 631, 3], [839, 2, 838, 3], [1, 0, 2, 1], [1, 2, 0, 1], [337, 2, 337, 3], [281, 2, 281, 3], [241, 2, 241, 3], [421, 2, 421, 3], [1, 1, 839, 0]];
sub<GL(2,Integers(840))|Gens>;
The image H : = ρ E ( Gal ( Q ‾ / Q ) ) H:=\rho_E(\Gal(\overline{\Q}/\Q)) H : = ρ E ( G a l ( Q / Q ) ) of the adelic Galois representation has
level 840 = 2 3 ⋅ 3 ⋅ 5 ⋅ 7 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 8 4 0 = 2 3 ⋅ 3 ⋅ 5 ⋅ 7 , index 2 2 2 , genus 0 0 0 , and generators
( 631 2 631 3 ) , ( 839 2 838 3 ) , ( 1 0 2 1 ) , ( 1 2 0 1 ) , ( 337 2 337 3 ) , ( 281 2 281 3 ) , ( 241 2 241 3 ) , ( 421 2 421 3 ) , ( 1 1 839 0 ) \left(\begin{array}{rr}
631 & 2 \\
631 & 3
\end{array}\right),\left(\begin{array}{rr}
839 & 2 \\
838 & 3
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
2 & 1
\end{array}\right),\left(\begin{array}{rr}
1 & 2 \\
0 & 1
\end{array}\right),\left(\begin{array}{rr}
337 & 2 \\
337 & 3
\end{array}\right),\left(\begin{array}{rr}
281 & 2 \\
281 & 3
\end{array}\right),\left(\begin{array}{rr}
241 & 2 \\
241 & 3
\end{array}\right),\left(\begin{array}{rr}
421 & 2 \\
421 & 3
\end{array}\right),\left(\begin{array}{rr}
1 & 1 \\
839 & 0
\end{array}\right) ( 6 3 1 6 3 1 2 3 ) , ( 8 3 9 8 3 8 2 3 ) , ( 1 2 0 1 ) , ( 1 0 2 1 ) , ( 3 3 7 3 3 7 2 3 ) , ( 2 8 1 2 8 1 2 3 ) , ( 2 4 1 2 4 1 2 3 ) , ( 4 2 1 4 2 1 2 3 ) , ( 1 8 3 9 1 0 ) .
The torsion field K : = Q ( E [ 840 ] ) K:=\Q(E[840]) K : = Q ( E [ 8 4 0 ] ) is a degree-35672555520 35672555520 3 5 6 7 2 5 5 5 5 2 0 Galois extension of Q \Q Q with Gal ( K / Q ) \Gal(K/\Q) G a l ( K / Q ) isomorphic to the projection of H H H to GL 2 ( Z / 840 Z ) \GL_2(\Z/840\Z) GL 2 ( Z / 8 4 0 Z ) .
The table below list all primes ℓ \ell ℓ for which the Serre invariants associated to the mod-ℓ \ell ℓ Galois representation are exceptional.
ℓ \ell ℓ
Reduction type
Serre weight
Serre conductor
2 2 2
nonsplit multiplicative
4 4 4
55545 = 3 ⋅ 5 ⋅ 7 ⋅ 2 3 2 55545 = 3 \cdot 5 \cdot 7 \cdot 23^{2} 5 5 5 4 5 = 3 ⋅ 5 ⋅ 7 ⋅ 2 3 2
3 3 3
nonsplit multiplicative
4 4 4
18515 = 5 ⋅ 7 ⋅ 2 3 2 18515 = 5 \cdot 7 \cdot 23^{2} 1 8 5 1 5 = 5 ⋅ 7 ⋅ 2 3 2
5 5 5
split multiplicative
6 6 6
22218 = 2 ⋅ 3 ⋅ 7 ⋅ 2 3 2 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2} 2 2 2 1 8 = 2 ⋅ 3 ⋅ 7 ⋅ 2 3 2
7 7 7
nonsplit multiplicative
8 8 8
15870 = 2 ⋅ 3 ⋅ 5 ⋅ 2 3 2 15870 = 2 \cdot 3 \cdot 5 \cdot 23^{2} 1 5 8 7 0 = 2 ⋅ 3 ⋅ 5 ⋅ 2 3 2
23 23 2 3
additive
112 112 1 1 2
210 = 2 ⋅ 3 ⋅ 5 ⋅ 7 210 = 2 \cdot 3 \cdot 5 \cdot 7 2 1 0 = 2 ⋅ 3 ⋅ 5 ⋅ 7
This curve has no rational isogenies. Its isogeny class 111090.l
consists of this curve only.
The minimal quadratic twist of this elliptic curve is
111090.c1 , its twist by − 23 -23 − 2 3 .
The number fields K K K of degree less than 24 such that
E ( K ) t o r s E(K)_{\rm tors} E ( K ) t o r s is strictly larger than E ( Q ) t o r s E(\Q)_{\rm tors} E ( Q ) t o r s
(which is trivial)
are as follows:
[ K : Q ] [K:\Q] [ K : Q ]
K K K
E ( K ) t o r s E(K)_{\rm tors} E ( K ) t o r s
Base change curve
3 3 3
3.1.444360.1
Z / 2 Z \Z/2\Z Z / 2 Z
not in database
6 6 6
6.0.165862880064000.1
Z / 2 Z ⊕ Z / 2 Z \Z/2\Z \oplus \Z/2\Z Z / 2 Z ⊕ Z / 2 Z
not in database
8 8 8
deg 8
Z / 3 Z \Z/3\Z Z / 3 Z
not in database
12 12 1 2
deg 12
Z / 4 Z \Z/4\Z Z / 4 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.
An entry - indicates that the invariants are not computed because the reduction is additive.
p p p -adic regulators
Note: p p p -adic regulator data only exists for primes p ≥ 5 p\ge 5 p ≥ 5 of good ordinary
reduction.
Choose a prime...
11
13
17
19
29
37
41
43
47
53
61
67
71
73
79
83
89
97