y 2 + x y = x 3 − 165713 x + 25950417 y^2+xy=x^3-165713x+25950417 y 2 + x y = x 3 − 1 6 5 7 1 3 x + 2 5 9 5 0 4 1 7
(homogenize , simplify )
y 2 z + x y z = x 3 − 165713 x z 2 + 25950417 z 3 y^2z+xyz=x^3-165713xz^2+25950417z^3 y 2 z + x y z = x 3 − 1 6 5 7 1 3 x z 2 + 2 5 9 5 0 4 1 7 z 3
(dehomogenize , simplify )
y 2 = x 3 − 214764075 x + 1211386947750 y^2=x^3-214764075x+1211386947750 y 2 = x 3 − 2 1 4 7 6 4 0 7 5 x + 1 2 1 1 3 8 6 9 4 7 7 5 0
(homogenize , minimize )
sage: E = EllipticCurve([1, 0, 0, -165713, 25950417])
gp: E = ellinit([1, 0, 0, -165713, 25950417])
magma: E := EllipticCurve([1, 0, 0, -165713, 25950417]);
oscar: E = elliptic_curve([1, 0, 0, -165713, 25950417])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Z ⊕ Z / 2 Z \Z \oplus \Z/{2}\Z Z ⊕ Z / 2 Z
magma: MordellWeilGroup(E);
P P P h ^ ( P ) \hat{h}(P) h ^ ( P ) Order
( 232 , − 41 ) (232, -41) ( 2 3 2 , − 4 1 ) 0.25454958251133535687088885620 0.25454958251133535687088885620 0 . 2 5 4 5 4 9 5 8 2 5 1 1 3 3 5 3 5 6 8 7 0 8 8 8 8 5 6 2 0 ∞ \infty ∞
( 943 / 4 , − 943 / 8 ) (943/4, -943/8) ( 9 4 3 / 4 , − 9 4 3 / 8 ) 0 0 0 2 2 2
( − 68 , 6109 ) \left(-68, 6109\right) ( − 6 8 , 6 1 0 9 ) , ( − 68 , − 6041 ) \left(-68, -6041\right) ( − 6 8 , − 6 0 4 1 ) , ( 232 , − 41 ) \left(232, -41\right) ( 2 3 2 , − 4 1 ) , ( 232 , − 191 ) \left(232, -191\right) ( 2 3 2 , − 1 9 1 ) , ( 256 , 439 ) \left(256, 439\right) ( 2 5 6 , 4 3 9 ) , ( 256 , − 695 ) \left(256, -695\right) ( 2 5 6 , − 6 9 5 ) , ( 292 , 1429 ) \left(292, 1429\right) ( 2 9 2 , 1 4 2 9 ) , ( 292 , − 1721 ) \left(292, -1721\right) ( 2 9 2 , − 1 7 2 1 ) , ( 2992 , 160729 ) \left(2992, 160729\right) ( 2 9 9 2 , 1 6 0 7 2 9 ) , ( 2992 , − 163721 ) \left(2992, -163721\right) ( 2 9 9 2 , − 1 6 3 7 2 1 )
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor :
N N N
=
4650 4650 4 6 5 0 = 2 ⋅ 3 ⋅ 5 2 ⋅ 31 2 \cdot 3 \cdot 5^{2} \cdot 31 2 ⋅ 3 ⋅ 5 2 ⋅ 3 1
sage: E.conductor().factor()
Discriminant :
Δ \Delta Δ
=
10296669375000 10296669375000 1 0 2 9 6 6 6 9 3 7 5 0 0 0 = 2 3 ⋅ 3 12 ⋅ 5 7 ⋅ 31 2^{3} \cdot 3^{12} \cdot 5^{7} \cdot 31 2 3 ⋅ 3 1 2 ⋅ 5 7 ⋅ 3 1
sage: E.discriminant().factor()
j-invariant :
j j j
=
32208729120020809 658986840 \frac{32208729120020809}{658986840} 6 5 8 9 8 6 8 4 0 3 2 2 0 8 7 2 9 1 2 0 0 2 0 8 0 9 = 2 − 3 ⋅ 3 − 12 ⋅ 5 − 1 ⋅ 3 1 − 1 ⋅ 37 3 3 ⋅ 85 3 3 2^{-3} \cdot 3^{-12} \cdot 5^{-1} \cdot 31^{-1} \cdot 373^{3} \cdot 853^{3} 2 − 3 ⋅ 3 − 1 2 ⋅ 5 − 1 ⋅ 3 1 − 1 ⋅ 3 7 3 3 ⋅ 8 5 3 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 ≈ 1.6164462287000309040940963952 1.6164462287000309040940963952 1 . 6 1 6 4 4 6 2 2 8 7 0 0 0 3 0 9 0 4 0 9 4 0 9 6 3 9 5 2
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 ≈ 0.81172727248298071679371672859 0.81172727248298071679371672859 0 . 8 1 1 7 2 7 2 7 2 4 8 2 9 8 0 7 1 6 7 9 3 7 1 6 7 2 8 5 9
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
a b c abc a b c quality :
Q Q Q ≈ 1.1107656438024482 1.1107656438024482 1 . 1 1 0 7 6 5 6 4 3 8 0 2 4 4 8 2
Szpiro ratio :
σ m \sigma_{m} σ m ≈ 5.644733246988683 5.644733246988683 5 . 6 4 4 7 3 3 2 4 6 9 8 8 6 8 3
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 ) ≈ 0.25454958251133535687088885620 0.25454958251133535687088885620 0 . 2 5 4 5 4 9 5 8 2 5 1 1 3 3 5 3 5 6 8 7 0 8 8 8 8 5 6 2 0
G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
Real period :
Ω \Omega Ω ≈ 0.66676333050773126183306416491 0.66676333050773126183306416491 0 . 6 6 6 7 6 3 3 3 0 5 0 7 7 3 1 2 6 1 8 3 3 0 6 4 1 6 4 9 1
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 = 144 144 1 4 4
= 3 ⋅ ( 2 2 ⋅ 3 ) ⋅ 2 2 ⋅ 1 3\cdot( 2^{2} \cdot 3 )\cdot2^{2}\cdot1 3 ⋅ ( 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 = 2 2 2
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
Special value :
L ′ ( E , 1 ) L'(E,1) L ′ ( E , 1 ) ≈ 6.1100757869259782167751598027 6.1100757869259782167751598027 6 . 1 1 0 0 7 5 7 8 6 9 2 5 9 7 8 2 1 6 7 7 5 1 5 9 8 0 2 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
(rounded )
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
6.110075787 ≈ L ′ ( E , 1 ) = # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p # E ( Q ) t o r 2 ≈ 1 ⋅ 0.666763 ⋅ 0.254550 ⋅ 144 2 2 ≈ 6.110075787 \displaystyle 6.110075787 \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 0.666763 \cdot 0.254550 \cdot 144}{2^2} \approx 6.110075787 6 . 1 1 0 0 7 5 7 8 7 ≈ L ′ ( E , 1 ) = # E ( Q ) t o r 2 # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p ≈ 2 2 1 ⋅ 0 . 6 6 6 7 6 3 ⋅ 0 . 2 5 4 5 5 0 ⋅ 1 4 4 ≈ 6 . 1 1 0 0 7 5 7 8 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
4650.2.a.bp
q + q 2 + q 3 + q 4 + q 6 + q 8 + q 9 − 4 q 11 + q 12 − 6 q 13 + q 16 − 2 q 17 + q 18 − 4 q 19 + O ( q 20 ) q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 4 q^{11} + q^{12} - 6 q^{13} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20}) q + q 2 + q 3 + q 4 + q 6 + q 8 + q 9 − 4 q 1 1 + q 1 2 − 6 q 1 3 + q 1 6 − 2 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 not semistable .
There
are 4 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], [1, 8, 0, 1], [1, 4, 4, 17], [3256, 1403, 2329, 2358], [2972, 3719, 2209, 3714], [3484, 1, 503, 6], [7, 6, 3714, 3715], [2328, 473, 2353, 2400], [3713, 8, 3712, 9], [1241, 8, 1244, 33]]
GL(2,Integers(3720)).subgroup(gens)
Gens := [[1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [3256, 1403, 2329, 2358], [2972, 3719, 2209, 3714], [3484, 1, 503, 6], [7, 6, 3714, 3715], [2328, 473, 2353, 2400], [3713, 8, 3712, 9], [1241, 8, 1244, 33]];
sub<GL(2,Integers(3720))|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 3720 = 2 3 ⋅ 3 ⋅ 5 ⋅ 31 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 3 7 2 0 = 2 3 ⋅ 3 ⋅ 5 ⋅ 3 1 , index 48 48 4 8 , genus 0 0 0 , and generators
( 1 0 8 1 ) , ( 1 8 0 1 ) , ( 1 4 4 17 ) , ( 3256 1403 2329 2358 ) , ( 2972 3719 2209 3714 ) , ( 3484 1 503 6 ) , ( 7 6 3714 3715 ) , ( 2328 473 2353 2400 ) , ( 3713 8 3712 9 ) , ( 1241 8 1244 33 ) \left(\begin{array}{rr}
1 & 0 \\
8 & 1
\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}
3256 & 1403 \\
2329 & 2358
\end{array}\right),\left(\begin{array}{rr}
2972 & 3719 \\
2209 & 3714
\end{array}\right),\left(\begin{array}{rr}
3484 & 1 \\
503 & 6
\end{array}\right),\left(\begin{array}{rr}
7 & 6 \\
3714 & 3715
\end{array}\right),\left(\begin{array}{rr}
2328 & 473 \\
2353 & 2400
\end{array}\right),\left(\begin{array}{rr}
3713 & 8 \\
3712 & 9
\end{array}\right),\left(\begin{array}{rr}
1241 & 8 \\
1244 & 33
\end{array}\right) ( 1 8 0 1 ) , ( 1 0 8 1 ) , ( 1 4 4 1 7 ) , ( 3 2 5 6 2 3 2 9 1 4 0 3 2 3 5 8 ) , ( 2 9 7 2 2 2 0 9 3 7 1 9 3 7 1 4 ) , ( 3 4 8 4 5 0 3 1 6 ) , ( 7 3 7 1 4 6 3 7 1 5 ) , ( 2 3 2 8 2 3 5 3 4 7 3 2 4 0 0 ) , ( 3 7 1 3 3 7 1 2 8 9 ) , ( 1 2 4 1 1 2 4 4 8 3 3 ) .
The torsion field K : = Q ( E [ 3720 ] ) K:=\Q(E[3720]) K : = Q ( E [ 3 7 2 0 ] ) is a degree-658243584000 658243584000 6 5 8 2 4 3 5 8 4 0 0 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 / 3720 Z ) \GL_2(\Z/3720\Z) GL 2 ( Z / 3 7 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
split multiplicative
4 4 4
775 = 5 2 ⋅ 31 775 = 5^{2} \cdot 31 7 7 5 = 5 2 ⋅ 3 1
3 3 3
split multiplicative
4 4 4
775 = 5 2 ⋅ 31 775 = 5^{2} \cdot 31 7 7 5 = 5 2 ⋅ 3 1
5 5 5
additive
18 18 1 8
186 = 2 ⋅ 3 ⋅ 31 186 = 2 \cdot 3 \cdot 31 1 8 6 = 2 ⋅ 3 ⋅ 3 1
31 31 3 1
nonsplit multiplicative
32 32 3 2
150 = 2 ⋅ 3 ⋅ 5 2 150 = 2 \cdot 3 \cdot 5^{2} 1 5 0 = 2 ⋅ 3 ⋅ 5 2
This curve has non-trivial cyclic isogenies of degree d d d for d = d= d =
2 and 4.
Its isogeny class 4650bi
consists of 4 curves linked by isogenies of
degrees dividing 4.
The minimal quadratic twist of this elliptic curve is
930a3 , its twist by 5 5 5 .
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 / 2 Z \cong \Z/{2}\Z ≅ Z / 2 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 ( 310 ) \Q(\sqrt{310}) Q ( 3 1 0 )
Z / 2 Z ⊕ Z / 2 Z \Z/2\Z \oplus \Z/2\Z Z / 2 Z ⊕ Z / 2 Z
not in database
2 2 2
Q ( 31 ) \Q(\sqrt{31}) Q ( 3 1 )
Z / 4 Z \Z/4\Z Z / 4 Z
not in database
2 2 2
Q ( 10 ) \Q(\sqrt{10}) Q ( 1 0 )
Z / 4 Z \Z/4\Z Z / 4 Z
not in database
4 4 4
Q ( 10 , 31 ) \Q(\sqrt{10}, \sqrt{31}) Q ( 1 0 , 3 1 )
Z / 2 Z ⊕ Z / 4 Z \Z/2\Z \oplus \Z/4\Z Z / 2 Z ⊕ Z / 4 Z
not in database
8 8 8
deg 8
Z / 2 Z ⊕ Z / 4 Z \Z/2\Z \oplus \Z/4\Z Z / 2 Z ⊕ Z / 4 Z
not in database
8 8 8
deg 8
Z / 8 Z \Z/8\Z Z / 8 Z
not in database
8 8 8
deg 8
Z / 8 Z \Z/8\Z Z / 8 Z
not in database
8 8 8
8.2.31558444171875.2
Z / 6 Z \Z/6\Z Z / 6 Z
not in database
16 16 1 6
deg 16
Z / 4 Z ⊕ Z / 4 Z \Z/4\Z \oplus \Z/4\Z Z / 4 Z ⊕ Z / 4 Z
not in database
16 16 1 6
deg 16
Z / 2 Z ⊕ Z / 8 Z \Z/2\Z \oplus \Z/8\Z Z / 2 Z ⊕ Z / 8 Z
not in database
16 16 1 6
deg 16
Z / 2 Z ⊕ Z / 8 Z \Z/2\Z \oplus \Z/8\Z Z / 2 Z ⊕ Z / 8 Z
not in database
16 16 1 6
deg 16
Z / 2 Z ⊕ Z / 6 Z \Z/2\Z \oplus \Z/6\Z Z / 2 Z ⊕ Z / 6 Z
not in database
16 16 1 6
deg 16
Z / 12 Z \Z/12\Z Z / 1 2 Z
not in database
16 16 1 6
deg 16
Z / 12 Z \Z/12\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.
An entry - indicates that the invariants are not computed because the reduction is additive.
p p p -adic regulators
p p p -adic regulators are not yet computed for curves that are not Γ 0 \Gamma_0 Γ 0 -optimal.