sage: E = EllipticCurve([1, 1, 1, 3, -5])
gp: E = ellinit([1, 1, 1, 3, -5])
magma: E := EllipticCurve([1, 1, 1, 3, -5]);
oscar: E = elliptic_curve([1, 1, 1, 3, -5])
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
( 5 , 10 ) (5, 10) ( 5 , 1 0 ) 0.15368275090910843817674487812 0.15368275090910843817674487812 0 . 1 5 3 6 8 2 7 5 0 9 0 9 1 0 8 4 3 8 1 7 6 7 4 4 8 7 8 1 2 ∞ \infty ∞
( 1 , 0 ) \left(1, 0\right) ( 1 , 0 ) , ( 1 , − 2 ) \left(1, -2\right) ( 1 , − 2 ) , ( 5 , 10 ) \left(5, 10\right) ( 5 , 1 0 ) , ( 5 , − 16 ) \left(5, -16\right) ( 5 , − 1 6 ) , ( 17 , 64 ) \left(17, 64\right) ( 1 7 , 6 4 ) , ( 17 , − 82 ) \left(17, -82\right) ( 1 7 , − 8 2 )
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor :
N N N
=
338 338 3 3 8 = 2 ⋅ 1 3 2 2 \cdot 13^{2} 2 ⋅ 1 3 2
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
Discriminant :
Δ \Delta Δ
=
− 17576 -17576 − 1 7 5 7 6 = − 1 ⋅ 2 3 ⋅ 1 3 3 -1 \cdot 2^{3} \cdot 13^{3} − 1 ⋅ 2 3 ⋅ 1 3 3
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
j-invariant :
j j j
=
1331 8 \frac{1331}{8} 8 1 3 3 1 = 2 − 3 ⋅ 1 1 3 2^{-3} \cdot 11^{3} 2 − 3 ⋅ 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 ≈ − 0.50357809758968124519938431369 -0.50357809758968124519938431369 − 0 . 5 0 3 5 7 8 0 9 7 5 8 9 6 8 1 2 4 5 1 9 9 3 8 4 3 1 3 6 9
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 ≈ − 1.1448154369550654292127561741 -1.1448154369550654292127561741 − 1 . 1 4 4 8 1 5 4 3 6 9 5 5 0 6 5 4 2 9 2 1 2 7 5 6 1 7 4 1
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
a b c abc a b c quality :
Q Q Q ≈ 0.9357683713670925 0.9357683713670925 0 . 9 3 5 7 6 8 3 7 1 3 6 7 0 9 2 5
Szpiro ratio :
σ m \sigma_{m} σ m ≈ 2.94137675475005 2.94137675475005 2 . 9 4 1 3 7 6 7 5 4 7 5 0 0 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 ) ≈ 0.15368275090910843817674487812 0.15368275090910843817674487812 0 . 1 5 3 6 8 2 7 5 0 9 0 9 1 0 8 4 3 8 1 7 6 7 4 4 8 7 8 1 2
sage: E.regulator()
gp: G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma: Regulator(E);
Real period :
Ω \Omega Ω ≈ 1.9456540408901337455759826091 1.9456540408901337455759826091 1 . 9 4 5 6 5 4 0 4 0 8 9 0 1 3 3 7 4 5 5 7 5 9 8 2 6 0 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 = 6 6 6
= 3 ⋅ 2 3\cdot2 3 ⋅ 2
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 ) ≈ 1.7940807919285122494029160801 1.7940807919285122494029160801 1 . 7 9 4 0 8 0 7 9 1 9 2 8 5 1 2 2 4 9 4 0 2 9 1 6 0 8 0 1
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);
1.794080792 ≈ L ′ ( E , 1 ) = # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p # E ( Q ) t o r 2 ≈ 1 ⋅ 1.945654 ⋅ 0.153683 ⋅ 6 1 2 ≈ 1.794080792 \begin{aligned} 1.794080792 \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 1.945654 \cdot 0.153683 \cdot 6}{1^2} \\ & \approx 1.794080792\end{aligned} 1 . 7 9 4 0 8 0 7 9 2 ≈ 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 ⋅ 1 . 9 4 5 6 5 4 ⋅ 0 . 1 5 3 6 8 3 ⋅ 6 ≈ 1 . 7 9 4 0 8 0 7 9 2
sage: # self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([1, 1, 1, 3, -5]); 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, 3, -5]); 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
338.2.a.d
q + q 2 − q 3 + q 4 − 3 q 5 − q 6 − 3 q 7 + q 8 − 2 q 9 − 3 q 10 − q 12 − 3 q 14 + 3 q 15 + q 16 − 3 q 17 − 2 q 18 − 6 q 19 + O ( q 20 ) q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9} - 3 q^{10} - q^{12} - 3 q^{14} + 3 q^{15} + q^{16} - 3 q^{17} - 2 q^{18} - 6 q^{19} + O(q^{20}) q + q 2 − q 3 + q 4 − 3 q 5 − q 6 − 3 q 7 + q 8 − 2 q 9 − 3 q 1 0 − q 1 2 − 3 q 1 4 + 3 q 1 5 + q 1 6 − 3 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 = [[481, 480, 1080, 481], [521, 0, 0, 521], [1351, 630, 0, 1351], [1227, 1397, 10, 873], [391, 150, 1185, 691], [1261, 30, 1290, 1297], [1, 1248, 240, 1], [1, 0, 1500, 1], [736, 1095, 1425, 151], [781, 0, 1500, 781], [521, 150, 0, 1], [1081, 0, 0, 241], [1, 0, 30, 1], [361, 1410, 750, 181], [16, 375, 765, 1306], [1, 0, 1080, 1]]
GL(2,Integers(1560)).subgroup(gens)
magma: Gens := [[481, 480, 1080, 481], [521, 0, 0, 521], [1351, 630, 0, 1351], [1227, 1397, 10, 873], [391, 150, 1185, 691], [1261, 30, 1290, 1297], [1, 1248, 240, 1], [1, 0, 1500, 1], [736, 1095, 1425, 151], [781, 0, 1500, 781], [521, 150, 0, 1], [1081, 0, 0, 241], [1, 0, 30, 1], [361, 1410, 750, 181], [16, 375, 765, 1306], [1, 0, 1080, 1]];
sub<GL(2,Integers(1560))|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 1560 = 2 3 ⋅ 3 ⋅ 5 ⋅ 13 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 1 5 6 0 = 2 3 ⋅ 3 ⋅ 5 ⋅ 1 3 , index 576 576 5 7 6 , genus 17 17 1 7 , and generators
( 481 480 1080 481 ) , ( 521 0 0 521 ) , ( 1351 630 0 1351 ) , ( 1227 1397 10 873 ) , ( 391 150 1185 691 ) , ( 1261 30 1290 1297 ) , ( 1 1248 240 1 ) , ( 1 0 1500 1 ) , ( 736 1095 1425 151 ) , ( 781 0 1500 781 ) , ( 521 150 0 1 ) , ( 1081 0 0 241 ) , ( 1 0 30 1 ) , ( 361 1410 750 181 ) , ( 16 375 765 1306 ) , ( 1 0 1080 1 ) \left(\begin{array}{rr}
481 & 480 \\
1080 & 481
\end{array}\right),\left(\begin{array}{rr}
521 & 0 \\
0 & 521
\end{array}\right),\left(\begin{array}{rr}
1351 & 630 \\
0 & 1351
\end{array}\right),\left(\begin{array}{rr}
1227 & 1397 \\
10 & 873
\end{array}\right),\left(\begin{array}{rr}
391 & 150 \\
1185 & 691
\end{array}\right),\left(\begin{array}{rr}
1261 & 30 \\
1290 & 1297
\end{array}\right),\left(\begin{array}{rr}
1 & 1248 \\
240 & 1
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
1500 & 1
\end{array}\right),\left(\begin{array}{rr}
736 & 1095 \\
1425 & 151
\end{array}\right),\left(\begin{array}{rr}
781 & 0 \\
1500 & 781
\end{array}\right),\left(\begin{array}{rr}
521 & 150 \\
0 & 1
\end{array}\right),\left(\begin{array}{rr}
1081 & 0 \\
0 & 241
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
30 & 1
\end{array}\right),\left(\begin{array}{rr}
361 & 1410 \\
750 & 181
\end{array}\right),\left(\begin{array}{rr}
16 & 375 \\
765 & 1306
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
1080 & 1
\end{array}\right) ( 4 8 1 1 0 8 0 4 8 0 4 8 1 ) , ( 5 2 1 0 0 5 2 1 ) , ( 1 3 5 1 0 6 3 0 1 3 5 1 ) , ( 1 2 2 7 1 0 1 3 9 7 8 7 3 ) , ( 3 9 1 1 1 8 5 1 5 0 6 9 1 ) , ( 1 2 6 1 1 2 9 0 3 0 1 2 9 7 ) , ( 1 2 4 0 1 2 4 8 1 ) , ( 1 1 5 0 0 0 1 ) , ( 7 3 6 1 4 2 5 1 0 9 5 1 5 1 ) , ( 7 8 1 1 5 0 0 0 7 8 1 ) , ( 5 2 1 0 1 5 0 1 ) , ( 1 0 8 1 0 0 2 4 1 ) , ( 1 3 0 0 1 ) , ( 3 6 1 7 5 0 1 4 1 0 1 8 1 ) , ( 1 6 7 6 5 3 7 5 1 3 0 6 ) , ( 1 1 0 8 0 0 1 ) .
The torsion field K : = Q ( E [ 1560 ] ) K:=\Q(E[1560]) K : = Q ( E [ 1 5 6 0 ] ) is a degree-1610219520 1610219520 1 6 1 0 2 1 9 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 / 1560 Z ) \GL_2(\Z/1560\Z) GL 2 ( Z / 1 5 6 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 =
5.
Its isogeny class 338.d
consists of 2 curves linked by isogenies of
degree 5.
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 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
7
11
23
29
31
43
53
59
61
67
71
73
79
83
89
97