y 2 + x y = x 3 + x 2 − 267192 x − 123117504 y^2+xy=x^3+x^2-267192x-123117504 y 2 + x y = x 3 + x 2 − 2 6 7 1 9 2 x − 1 2 3 1 1 7 5 0 4
(homogenize , simplify )
y 2 z + x y z = x 3 + x 2 z − 267192 x z 2 − 123117504 z 3 y^2z+xyz=x^3+x^2z-267192xz^2-123117504z^3 y 2 z + x y z = x 3 + x 2 z − 2 6 7 1 9 2 x z 2 − 1 2 3 1 1 7 5 0 4 z 3
(dehomogenize , simplify )
y 2 = x 3 − 346281507 x − 5738976047394 y^2=x^3-346281507x-5738976047394 y 2 = x 3 − 3 4 6 2 8 1 5 0 7 x − 5 7 3 8 9 7 6 0 4 7 3 9 4
(homogenize , minimize )
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)
Z / 4 Z \Z/{4}\Z Z / 4 Z
magma: MordellWeilGroup(E);
( 1712 , 65784 ) \left(1712, 65784\right) ( 1 7 1 2 , 6 5 7 8 4 ) , ( 1712 , − 67496 ) \left(1712, -67496\right) ( 1 7 1 2 , − 6 7 4 9 6 )
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor :
N N N
=
82110 82110 8 2 1 1 0 = 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 17 ⋅ 23 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 1 7 ⋅ 2 3
sage: E.conductor().factor()
Discriminant :
Δ \Delta Δ
=
− 5315560665303840000 -5315560665303840000 − 5 3 1 5 5 6 0 6 6 5 3 0 3 8 4 0 0 0 0 = − 1 ⋅ 2 8 ⋅ 3 ⋅ 5 4 ⋅ 7 8 ⋅ 1 7 4 ⋅ 23 -1 \cdot 2^{8} \cdot 3 \cdot 5^{4} \cdot 7^{8} \cdot 17^{4} \cdot 23 − 1 ⋅ 2 8 ⋅ 3 ⋅ 5 4 ⋅ 7 8 ⋅ 1 7 4 ⋅ 2 3
sage: E.discriminant().factor()
j-invariant :
j j j
=
− 2109582937351555472521 5315560665303840000 -\frac{2109582937351555472521}{5315560665303840000} − 5 3 1 5 5 6 0 6 6 5 3 0 3 8 4 0 0 0 0 2 1 0 9 5 8 2 9 3 7 3 5 1 5 5 5 4 7 2 5 2 1 = − 1 ⋅ 2 − 8 ⋅ 3 − 1 ⋅ 5 − 4 ⋅ 7 − 8 ⋅ 1 1 3 ⋅ 1 3 6 ⋅ 1 7 − 4 ⋅ 2 3 − 1 ⋅ 689 9 3 -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} − 1 ⋅ 2 − 8 ⋅ 3 − 1 ⋅ 5 − 4 ⋅ 7 − 8 ⋅ 1 1 3 ⋅ 1 3 6 ⋅ 1 7 − 4 ⋅ 2 3 − 1 ⋅ 6 8 9 9 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 ≈ 2.2809353407557665373004355108 2.2809353407557665373004355108 2 . 2 8 0 9 3 5 3 4 0 7 5 5 7 6 6 5 3 7 3 0 0 4 3 5 5 1 0 8
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 ≈ 2.2809353407557665373004355108 2.2809353407557665373004355108 2 . 2 8 0 9 3 5 3 4 0 7 5 5 7 6 6 5 3 7 3 0 0 4 3 5 5 1 0 8
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
a b c abc a b c quality :
Q Q Q ≈ 1.0053760035927872 1.0053760035927872 1 . 0 0 5 3 7 6 0 0 3 5 9 2 7 8 7 2
Szpiro ratio :
σ m \sigma_{m} σ m ≈ 4.487404163344251 4.487404163344251 4 . 4 8 7 4 0 4 1 6 3 3 4 4 2 5 1
Analytic rank :
r a n r_{\mathrm{an}} r a n = 0 0 0
Mordell-Weil rank :
r r r = 0 0 0
gp: [lower,upper] = ellrank(E)
Regulator :
R e g ( E / Q ) \mathrm{Reg}(E/\Q) R e g ( E / Q ) = 1 1 1
G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
Real period :
Ω \Omega Ω ≈ 0.097765187202809812022863587980 0.097765187202809812022863587980 0 . 0 9 7 7 6 5 1 8 7 2 0 2 8 0 9 8 1 2 0 2 2 8 6 3 5 8 7 9 8 0
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 = 256 256 2 5 6
= 2 ⋅ 1 ⋅ 2 2 ⋅ 2 3 ⋅ 2 2 ⋅ 1 2\cdot1\cdot2^{2}\cdot2^{3}\cdot2^{2}\cdot1 2 ⋅ 1 ⋅ 2 2 ⋅ 2 3 ⋅ 2 2 ⋅ 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 = 4 4 4
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
Special value :
L ( E , 1 ) L(E,1) L ( E , 1 ) ≈ 1.5642429952449569923658174077 1.5642429952449569923658174077 1 . 5 6 4 2 4 2 9 9 5 2 4 4 9 5 6 9 9 2 3 6 5 8 1 7 4 0 7 7
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
(exact )
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
1.564242995 ≈ L ( E , 1 ) = # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p # E ( Q ) t o r 2 ≈ 1 ⋅ 0.097765 ⋅ 1.000000 ⋅ 256 4 2 ≈ 1.564242995 \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 1 . 5 6 4 2 4 2 9 9 5 ≈ L ( E , 1 ) = # E ( Q ) t o r 2 # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p ≈ 4 2 1 ⋅ 0 . 0 9 7 7 6 5 ⋅ 1 . 0 0 0 0 0 0 ⋅ 2 5 6 ≈ 1 . 5 6 4 2 4 2 9 9 5
# 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
82110.2.a.o
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 ) 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}) q − q 2 − q 3 + q 4 + q 5 + q 6 + q 7 − q 8 + q 9 − q 1 0 − q 1 2 − 2 q 1 3 − q 1 4 − q 1 5 + q 1 6 + q 1 7 − q 1 8 + 4 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 semistable .
There
are 6 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 except those listed in the table below.
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 : = 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 46920 = 2 3 ⋅ 3 ⋅ 5 ⋅ 17 ⋅ 23 46920 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \cdot 23 4 6 9 2 0 = 2 3 ⋅ 3 ⋅ 5 ⋅ 1 7 ⋅ 2 3 , index 48 48 4 8 , genus 0 0 0 , and generators
( 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 ) \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) ( 1 8 0 1 ) , ( 2 9 3 2 8 2 9 3 5 3 5 8 7 3 2 9 4 0 0 ) , ( 1 0 8 1 ) , ( 1 4 4 1 7 ) , ( 4 1 0 5 9 2 9 3 3 8 4 1 0 5 8 5 8 7 5 ) , ( 2 0 4 4 2 3 1 6 ) , ( 1 6 5 6 1 1 9 3 2 4 8 3 3 ) , ( 3 7 5 3 7 9 3 8 8 8 3 3 ) , ( 7 4 6 9 1 4 6 4 6 9 1 5 ) , ( 1 5 6 4 4 3 1 3 0 3 1 6 ) , ( 4 6 9 1 3 4 6 9 1 2 8 9 ) .
The torsion field K : = Q ( E [ 46920 ] ) K:=\Q(E[46920]) K : = Q ( E [ 4 6 9 2 0 ] ) is a degree-15430439078461440 15430439078461440 1 5 4 3 0 4 3 9 0 7 8 4 6 1 4 4 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 / 46920 Z ) \GL_2(\Z/46920\Z) GL 2 ( Z / 4 6 9 2 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
69 = 3 ⋅ 23 69 = 3 \cdot 23 6 9 = 3 ⋅ 2 3
3 3 3
nonsplit multiplicative
4 4 4
27370 = 2 ⋅ 5 ⋅ 7 ⋅ 17 ⋅ 23 27370 = 2 \cdot 5 \cdot 7 \cdot 17 \cdot 23 2 7 3 7 0 = 2 ⋅ 5 ⋅ 7 ⋅ 1 7 ⋅ 2 3
5 5 5
split multiplicative
6 6 6
16422 = 2 ⋅ 3 ⋅ 7 ⋅ 17 ⋅ 23 16422 = 2 \cdot 3 \cdot 7 \cdot 17 \cdot 23 1 6 4 2 2 = 2 ⋅ 3 ⋅ 7 ⋅ 1 7 ⋅ 2 3
7 7 7
split multiplicative
8 8 8
11730 = 2 ⋅ 3 ⋅ 5 ⋅ 17 ⋅ 23 11730 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 23 1 1 7 3 0 = 2 ⋅ 3 ⋅ 5 ⋅ 1 7 ⋅ 2 3
17 17 1 7
split multiplicative
18 18 1 8
4830 = 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 23 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 4 8 3 0 = 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 2 3
23 23 2 3
nonsplit multiplicative
24 24 2 4
3570 = 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 17 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 3 5 7 0 = 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 1 7
This curve has non-trivial cyclic isogenies of degree d d d for d = d= d =
2 and 4.
Its isogeny class 82110.o
consists of 4 curves linked by isogenies of
degrees dividing 4.
This elliptic curve is its own minimal quadratic twist .
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
≅ Z / 4 Z \cong \Z/{4}\Z ≅ Z / 4 Z
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
2 2 2
Q ( − 69 ) \Q(\sqrt{-69}) Q ( − 6 9 )
Z / 2 Z ⊕ Z / 4 Z \Z/2\Z \oplus \Z/4\Z Z / 2 Z ⊕ Z / 4 Z
not in database
4 4 4
4.2.7976400.8
Z / 8 Z \Z/8\Z Z / 8 Z
not in database
8 8 8
8.0.7072524735676416.8
Z / 4 Z ⊕ Z / 4 Z \Z/4\Z \oplus \Z/4\Z Z / 4 Z ⊕ Z / 4 Z
not in database
8 8 8
deg 8
Z / 2 Z ⊕ Z / 8 Z \Z/2\Z \oplus \Z/8\Z Z / 2 Z ⊕ Z / 8 Z
not in database
8 8 8
deg 8
Z / 2 Z ⊕ Z / 8 Z \Z/2\Z \oplus \Z/8\Z Z / 2 Z ⊕ Z / 8 Z
not in database
8 8 8
deg 8
Z / 12 Z \Z/12\Z Z / 1 2 Z
not in database
16 16 1 6
deg 16
Z / 16 Z \Z/16\Z Z / 1 6 Z
not in database
16 16 1 6
deg 16
Z / 2 Z ⊕ Z / 12 Z \Z/2\Z \oplus \Z/12\Z Z / 2 Z ⊕ Z / 1 2 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.
All Iwasawa λ \lambda λ and μ \mu μ -invariants for primes p ≥ 3 p\ge
3 p ≥ 3 of good reduction are zero.
p p p -adic regulators
All p p p -adic regulators are identically 1 1 1 since the rank is 0 0 0 .