Minimal Weierstrass equation
sage: E = EllipticCurve([1, -1, 0, -11964633, 15936473069])
gp: E = ellinit([1, -1, 0, -11964633, 15936473069])
magma: E := EllipticCurve([1, -1, 0, -11964633, 15936473069]);
Minimal equation
Minimal equation
Simplified equation
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\(y^2+xy=x^3-x^2-11964633x+15936473069\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-11964633xz^2+15936473069z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-191434131x+1019742842286\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 9702 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 11$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-57307978910298650688 $ | = | $-1 \cdot 2^{6} \cdot 3^{9} \cdot 7^{10} \cdot 11^{5} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{34068278205171}{10307264} \) | = | $-1 \cdot 2^{-6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{-5} \cdot 2953^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $2.7688462006097401859222322511\dots$ | ||
| Stable Faltings height: | $0.32329519322923016312133770387\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.19393398146725376875331045033\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | $ 4 $ = $ 2\cdot2\cdot1\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $1$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L(E,1) $ ≈ $ 0.77573592586901507501324180132 $ | ||
Modular invariants
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
magma: ModularForm(E);
\( q - q^{2} + q^{4} - 2 q^{5} - q^{8} + 2 q^{10} - q^{11} - 2 q^{13} + q^{16} + q^{17} + 3 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 423360 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 0 |
| $7$ | $1$ | $II^{*}$ | Additive | -1 | 2 | 10 | 0 |
| $11$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
Galois representations
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
$p$-adic regulators
sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | ord | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | 0 | - | 0 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 0 |
| $\mu$-invariant(s) | 0 | - | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has no rational isogenies. Its isogeny class 9702.f consists of this curve only.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.6468.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.5522223168.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $8$ | 8.2.3767105332683.2 | \(\Z/3\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | 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.