sage: E = EllipticCurve([1, 1, 1, -8, 6])
gp: E = ellinit([1, 1, 1, -8, 6])
magma: E := EllipticCurve([1, 1, 1, -8, 6]);
oscar: E = elliptic_curve([1, 1, 1, -8, 6])
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
( 0 , 2 ) (0, 2) ( 0 , 2 ) 0.17505957779114896685301051798 0.17505957779114896685301051798 0 . 1 7 5 0 5 9 5 7 7 7 9 1 1 4 8 9 6 6 8 5 3 0 1 0 5 1 7 9 8 ∞ \infty ∞
( 0 , 2 ) \left(0, 2\right) ( 0 , 2 ) , ( 0 , − 3 ) \left(0, -3\right) ( 0 , − 3 ) , ( 1 , 0 ) \left(1, 0\right) ( 1 , 0 ) , ( 1 , − 2 ) \left(1, -2\right) ( 1 , − 2 ) , ( 10 , 27 ) \left(10, 27\right) ( 1 0 , 2 7 ) , ( 10 , − 38 ) \left(10, -38\right) ( 1 0 , − 3 8 )
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor :
N N N
=
1225 1225 1 2 2 5 = 5 2 ⋅ 7 2 5^{2} \cdot 7^{2} 5 2 ⋅ 7 2
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
Discriminant :
Δ \Delta Δ
=
− 6125 -6125 − 6 1 2 5 = − 1 ⋅ 5 3 ⋅ 7 2 -1 \cdot 5^{3} \cdot 7^{2} − 1 ⋅ 5 3 ⋅ 7 2
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
j-invariant :
j j j
=
− 9317 -9317 − 9 3 1 7 = − 1 ⋅ 7 ⋅ 1 1 3 -1 \cdot 7 \cdot 11^{3} − 1 ⋅ 7 ⋅ 1 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 ≈ − 0.53497422781474375918341356667 -0.53497422781474375918341356667 − 0 . 5 3 4 9 7 4 2 2 7 8 1 4 7 4 3 7 5 9 1 8 3 4 1 3 5 6 6 6 7
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 ≈ − 1.2616520640991544036844955239 -1.2616520640991544036844955239 − 1 . 2 6 1 6 5 2 0 6 4 0 9 9 1 5 4 4 0 3 6 8 4 4 9 5 5 2 3 9
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
a b c abc a b c quality :
Q Q Q ≈ 0.8028988663793745 0.8028988663793745 0 . 8 0 2 8 9 8 8 6 6 3 7 9 3 7 4 5
Szpiro ratio :
σ m \sigma_{m} σ m ≈ 2.5355980398321445 2.5355980398321445 2 . 5 3 5 5 9 8 0 3 9 8 3 2 1 4 4 5
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 ) ≈ 0.17505957779114896685301051798 0.17505957779114896685301051798 0 . 1 7 5 0 5 9 5 7 7 7 9 1 1 4 8 9 6 6 8 5 3 0 1 0 5 1 7 9 8
sage: E.regulator()
gp: G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma: Regulator(E);
Real period :
Ω \Omega Ω ≈ 4.1353485680374347938775159781 4.1353485680374347938775159781 4 . 1 3 5 3 4 8 5 6 8 0 3 7 4 3 4 7 9 3 8 7 7 5 1 5 9 7 8 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 ) ≈ 1.4478647486797316047223462949 1.4478647486797316047223462949 1 . 4 4 7 8 6 4 7 4 8 6 7 9 7 3 1 6 0 4 7 2 2 3 4 6 2 9 4 9
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);
1.447864749 ≈ L ′ ( E , 1 ) = # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p # E ( Q ) t o r 2 ≈ 1 ⋅ 4.135349 ⋅ 0.175060 ⋅ 2 1 2 ≈ 1.447864749 \begin{aligned} 1.447864749 \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 4.135349 \cdot 0.175060 \cdot 2}{1^2} \\ & \approx 1.447864749\end{aligned} 1 . 4 4 7 8 6 4 7 4 9 ≈ 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 ⋅ 4 . 1 3 5 3 4 9 ⋅ 0 . 1 7 5 0 6 0 ⋅ 2 ≈ 1 . 4 4 7 8 6 4 7 4 9
sage: # self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([1, 1, 1, -8, 6]); 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, 1, -8, 6]); 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
1225.2.a.b
q − q 2 − q 3 − q 4 + q 6 + 3 q 8 − 2 q 9 + q 12 + 2 q 13 − q 16 + 2 q 17 + 2 q 18 − 6 q 19 + O ( q 20 ) q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} - 2 q^{9} + q^{12} + 2 q^{13} - q^{16} + 2 q^{17} + 2 q^{18} - 6 q^{19} + O(q^{20}) q − q 2 − q 3 − q 4 + q 6 + 3 q 8 − 2 q 9 + q 1 2 + 2 q 1 3 − q 1 6 + 2 q 1 7 + 2 q 1 8 − 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 = [[101, 2072, 148, 2157], [1, 3256, 0, 1], [1, 0, 148, 1], [117, 2442, 37, 4323], [1, 0, 148, 3221], [1037, 0, 0, 2073], [1925, 3256, 1924, 1925], [4441, 0, 0, 3701], [1, 30, 0, 1], [75, 74, 1591, 2295], [1, 2590, 0, 1], [2591, 2590, 2590, 2591], [4848, 2923, 4477, 1259], [1, 0, 2590, 1]]
GL(2,Integers(5180)).subgroup(gens)
magma: Gens := [[101, 2072, 148, 2157], [1, 3256, 0, 1], [1, 0, 148, 1], [117, 2442, 37, 4323], [1, 0, 148, 3221], [1037, 0, 0, 2073], [1925, 3256, 1924, 1925], [4441, 0, 0, 3701], [1, 30, 0, 1], [75, 74, 1591, 2295], [1, 2590, 0, 1], [2591, 2590, 2590, 2591], [4848, 2923, 4477, 1259], [1, 0, 2590, 1]];
sub<GL(2,Integers(5180))|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 5180 = 2 2 ⋅ 5 ⋅ 7 ⋅ 37 5180 = 2^{2} \cdot 5 \cdot 7 \cdot 37 5 1 8 0 = 2 2 ⋅ 5 ⋅ 7 ⋅ 3 7 , index 2736 2736 2 7 3 6 , genus 97 97 9 7 , and generators
( 101 2072 148 2157 ) , ( 1 3256 0 1 ) , ( 1 0 148 1 ) , ( 117 2442 37 4323 ) , ( 1 0 148 3221 ) , ( 1037 0 0 2073 ) , ( 1925 3256 1924 1925 ) , ( 4441 0 0 3701 ) , ( 1 30 0 1 ) , ( 75 74 1591 2295 ) , ( 1 2590 0 1 ) , ( 2591 2590 2590 2591 ) , ( 4848 2923 4477 1259 ) , ( 1 0 2590 1 ) \left(\begin{array}{rr}
101 & 2072 \\
148 & 2157
\end{array}\right),\left(\begin{array}{rr}
1 & 3256 \\
0 & 1
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
148 & 1
\end{array}\right),\left(\begin{array}{rr}
117 & 2442 \\
37 & 4323
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
148 & 3221
\end{array}\right),\left(\begin{array}{rr}
1037 & 0 \\
0 & 2073
\end{array}\right),\left(\begin{array}{rr}
1925 & 3256 \\
1924 & 1925
\end{array}\right),\left(\begin{array}{rr}
4441 & 0 \\
0 & 3701
\end{array}\right),\left(\begin{array}{rr}
1 & 30 \\
0 & 1
\end{array}\right),\left(\begin{array}{rr}
75 & 74 \\
1591 & 2295
\end{array}\right),\left(\begin{array}{rr}
1 & 2590 \\
0 & 1
\end{array}\right),\left(\begin{array}{rr}
2591 & 2590 \\
2590 & 2591
\end{array}\right),\left(\begin{array}{rr}
4848 & 2923 \\
4477 & 1259
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
2590 & 1
\end{array}\right) ( 1 0 1 1 4 8 2 0 7 2 2 1 5 7 ) , ( 1 0 3 2 5 6 1 ) , ( 1 1 4 8 0 1 ) , ( 1 1 7 3 7 2 4 4 2 4 3 2 3 ) , ( 1 1 4 8 0 3 2 2 1 ) , ( 1 0 3 7 0 0 2 0 7 3 ) , ( 1 9 2 5 1 9 2 4 3 2 5 6 1 9 2 5 ) , ( 4 4 4 1 0 0 3 7 0 1 ) , ( 1 0 3 0 1 ) , ( 7 5 1 5 9 1 7 4 2 2 9 5 ) , ( 1 0 2 5 9 0 1 ) , ( 2 5 9 1 2 5 9 0 2 5 9 0 2 5 9 1 ) , ( 4 8 4 8 4 4 7 7 2 9 2 3 1 2 5 9 ) , ( 1 2 5 9 0 0 1 ) .
The torsion field K : = Q ( E [ 5180 ] ) K:=\Q(E[5180]) K : = Q ( E [ 5 1 8 0 ] ) is a degree-61869588480 61869588480 6 1 8 6 9 5 8 8 4 8 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 / 5180 Z ) \GL_2(\Z/5180\Z) GL 2 ( Z / 5 1 8 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.
gp: ellisomat(E)
This curve has non-trivial cyclic isogenies of degree d d d for d = d= d =
37.
Its isogeny class 1225.b
consists of 2 curves linked by isogenies of
degree 37.
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
(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...
19
23
29
31
37
41
43
47
53
61
67
71
73
79
83
89
97