y 2 + x y = x 3 − x 2 − 270 x + 1777 y^2+xy=x^3-x^2-270x+1777 y 2 + x y = x 3 − x 2 − 2 7 0 x + 1 7 7 7
(homogenize , simplify )
y 2 z + x y z = x 3 − x 2 z − 270 x z 2 + 1777 z 3 y^2z+xyz=x^3-x^2z-270xz^2+1777z^3 y 2 z + x y z = x 3 − x 2 z − 2 7 0 x z 2 + 1 7 7 7 z 3
(dehomogenize , simplify )
y 2 = x 3 − 4323 x + 109406 y^2=x^3-4323x+109406 y 2 = x 3 − 4 3 2 3 x + 1 0 9 4 0 6
(homogenize , minimize )
sage: E = EllipticCurve([1, -1, 0, -270, 1777])
gp: E = ellinit([1, -1, 0, -270, 1777])
magma: E := EllipticCurve([1, -1, 0, -270, 1777]);
oscar: E = elliptic_curve([1, -1, 0, -270, 1777])
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
( 8 , 5 ) (8, 5) ( 8 , 5 ) 0.46709978364418083706304954866 0.46709978364418083706304954866 0 . 4 6 7 0 9 9 7 8 3 6 4 4 1 8 0 8 3 7 0 6 3 0 4 9 5 4 8 6 6 ∞ \infty ∞
( 8 , 5 ) \left(8, 5\right) ( 8 , 5 ) , ( 8 , − 13 ) \left(8, -13\right) ( 8 , − 1 3 )
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor :
N N N
=
1089 1089 1 0 8 9 = 3 2 ⋅ 1 1 2 3^{2} \cdot 11^{2} 3 2 ⋅ 1 1 2
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
Discriminant :
Δ \Delta Δ
=
− 88209 -88209 − 8 8 2 0 9 = − 1 ⋅ 3 6 ⋅ 1 1 2 -1 \cdot 3^{6} \cdot 11^{2} − 1 ⋅ 3 6 ⋅ 1 1 2
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
j-invariant :
j j j
=
− 24729001 -24729001 − 2 4 7 2 9 0 0 1 = − 1 ⋅ 11 ⋅ 13 1 3 -1 \cdot 11 \cdot 131^{3} − 1 ⋅ 1 1 ⋅ 1 3 1 3
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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 )
sage: E.has_cm()
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 ≈ 0.031345306218375756312683160611 0.031345306218375756312683160611 0 . 0 3 1 3 4 5 3 0 6 2 1 8 3 7 5 7 5 6 3 1 2 6 8 3 1 6 0 6 1 1
gp: ellheight(E)
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 ≈ − 0.91761005024874084672859672084 -0.91761005024874084672859672084 − 0 . 9 1 7 6 1 0 0 5 0 2 4 8 7 4 0 8 4 6 7 2 8 5 9 6 7 2 0 8 4
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
a b c abc a b c quality :
Q Q Q ≈ 0.9685335876741098 0.9685335876741098 0 . 9 6 8 5 3 3 5 8 7 6 7 4 1 0 9 8
Szpiro ratio :
σ m \sigma_{m} σ m ≈ 4.062771336195671 4.062771336195671 4 . 0 6 2 7 7 1 3 3 6 1 9 5 6 7 1
Analytic rank :
r a n r_{\mathrm{an}} r a n = 1 1 1
sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
Mordell-Weil rank :
r r r = 1 1 1
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
Regulator :
R e g ( E / Q ) \mathrm{Reg}(E/\Q) R e g ( E / Q ) ≈ 0.46709978364418083706304954866 0.46709978364418083706304954866 0 . 4 6 7 0 9 9 7 8 3 6 4 4 1 8 0 8 3 7 0 6 3 0 4 9 5 4 8 6 6
sage: E.regulator()
gp: G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma: Regulator(E);
Real period :
Ω \Omega Ω ≈ 3.1904476027224831732295847251 3.1904476027224831732295847251 3 . 1 9 0 4 4 7 6 0 2 7 2 2 4 8 3 1 7 3 2 2 9 5 8 4 7 2 5 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 = 2 2 2
= 2 ⋅ 1 2\cdot1 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 = 1 1 1
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) L ′ ( E , 1 ) ≈ 2.9805147699195346135285188098 2.9805147699195346135285188098 2 . 9 8 0 5 1 4 7 6 9 9 1 9 5 3 4 6 1 3 5 2 8 5 1 8 8 0 9 8
sage: 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);
2.980514770 ≈ L ′ ( E , 1 ) = # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p # E ( Q ) t o r 2 ≈ 1 ⋅ 3.190448 ⋅ 0.467100 ⋅ 2 1 2 ≈ 2.980514770 \begin{aligned} 2.980514770 \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 3.190448 \cdot 0.467100 \cdot 2}{1^2} \\ & \approx 2.980514770\end{aligned} 2 . 9 8 0 5 1 4 7 7 0 ≈ 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 ⋅ 3 . 1 9 0 4 4 8 ⋅ 0 . 4 6 7 1 0 0 ⋅ 2 ≈ 2 . 9 8 0 5 1 4 7 7 0
sage: # self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([1, -1, 0, -270, 1777]); 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)))
magma: /* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
E := EllipticCurve([1, -1, 0, -270, 1777]); 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
1089.2.a.i
q + q 2 − q 4 − q 5 + 2 q 7 − 3 q 8 − q 10 − q 13 + 2 q 14 − q 16 − 5 q 17 − 6 q 19 + O ( q 20 ) q + q^{2} - q^{4} - q^{5} + 2 q^{7} - 3 q^{8} - q^{10} - q^{13} + 2 q^{14} - q^{16} - 5 q^{17} - 6 q^{19} + O(q^{20}) q + q 2 − q 4 − q 5 + 2 q 7 − 3 q 8 − q 1 0 − q 1 3 + 2 q 1 4 − q 1 6 − 5 q 1 7 − 6 q 1 9 + O ( q 2 0 )
sage: E.q_eigenform(20)
gp: \\ 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.
This elliptic curve is not semistable .
There
are 2 primes p p p
of bad reduction :
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)]
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)];
sage: gens = [[177, 88, 176, 177], [133, 132, 66, 1], [175, 0, 0, 263], [133, 33, 0, 67], [1, 108, 0, 1], [193, 132, 66, 169], [133, 132, 132, 133], [1, 132, 0, 1], [1, 0, 88, 1], [1, 0, 132, 1], [100, 231, 165, 199], [133, 198, 198, 1]]
GL(2,Integers(264)).subgroup(gens)
magma: Gens := [[177, 88, 176, 177], [133, 132, 66, 1], [175, 0, 0, 263], [133, 33, 0, 67], [1, 108, 0, 1], [193, 132, 66, 169], [133, 132, 132, 133], [1, 132, 0, 1], [1, 0, 88, 1], [1, 0, 132, 1], [100, 231, 165, 199], [133, 198, 198, 1]];
sub<GL(2,Integers(264))|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 264 = 2 3 ⋅ 3 ⋅ 11 264 = 2^{3} \cdot 3 \cdot 11 2 6 4 = 2 3 ⋅ 3 ⋅ 1 1 , index 480 480 4 8 0 , genus 16 16 1 6 , and generators
( 177 88 176 177 ) , ( 133 132 66 1 ) , ( 175 0 0 263 ) , ( 133 33 0 67 ) , ( 1 108 0 1 ) , ( 193 132 66 169 ) , ( 133 132 132 133 ) , ( 1 132 0 1 ) , ( 1 0 88 1 ) , ( 1 0 132 1 ) , ( 100 231 165 199 ) , ( 133 198 198 1 ) \left(\begin{array}{rr}
177 & 88 \\
176 & 177
\end{array}\right),\left(\begin{array}{rr}
133 & 132 \\
66 & 1
\end{array}\right),\left(\begin{array}{rr}
175 & 0 \\
0 & 263
\end{array}\right),\left(\begin{array}{rr}
133 & 33 \\
0 & 67
\end{array}\right),\left(\begin{array}{rr}
1 & 108 \\
0 & 1
\end{array}\right),\left(\begin{array}{rr}
193 & 132 \\
66 & 169
\end{array}\right),\left(\begin{array}{rr}
133 & 132 \\
132 & 133
\end{array}\right),\left(\begin{array}{rr}
1 & 132 \\
0 & 1
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
88 & 1
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
132 & 1
\end{array}\right),\left(\begin{array}{rr}
100 & 231 \\
165 & 199
\end{array}\right),\left(\begin{array}{rr}
133 & 198 \\
198 & 1
\end{array}\right) ( 1 7 7 1 7 6 8 8 1 7 7 ) , ( 1 3 3 6 6 1 3 2 1 ) , ( 1 7 5 0 0 2 6 3 ) , ( 1 3 3 0 3 3 6 7 ) , ( 1 0 1 0 8 1 ) , ( 1 9 3 6 6 1 3 2 1 6 9 ) , ( 1 3 3 1 3 2 1 3 2 1 3 3 ) , ( 1 0 1 3 2 1 ) , ( 1 8 8 0 1 ) , ( 1 1 3 2 0 1 ) , ( 1 0 0 1 6 5 2 3 1 1 9 9 ) , ( 1 3 3 1 9 8 1 9 8 1 ) .
The torsion field K : = Q ( E [ 264 ] ) K:=\Q(E[264]) K : = Q ( E [ 2 6 4 ] ) is a degree-2027520 2027520 2 0 2 7 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 / 264 Z ) \GL_2(\Z/264\Z) GL 2 ( Z / 2 6 4 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.
gp: ellisomat(E)
This curve has non-trivial cyclic isogenies of degree d d d for d = d= d =
11.
Its isogeny class 1089.i
consists of 2 curves linked by isogenies of
degree 11.
The minimal quadratic twist of this elliptic curve is
121.a2 , its twist by − 3 -3 − 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:
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 have not yet been computed.
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...
5
13
19
23
29
31
37
41
43
47
59
61
67
71
73
79
83
89
97