Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-27x-249\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-27xz^2-249z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-35019x-11512314\)
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(homogenize, minimize) |
sage: E = EllipticCurve([1, 0, 0, -27, -249])
gp: E = ellinit([1, 0, 0, -27, -249])
magma: E := EllipticCurve([1, 0, 0, -27, -249]);
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
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: | \( 966 \) | = | $2 \cdot 3 \cdot 7 \cdot 23$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-25039686 $ | = | $-1 \cdot 2 \cdot 3 \cdot 7^{3} \cdot 23^{3} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{2181825073}{25039686} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 7^{-3} \cdot 23^{-3} \cdot 1297^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.10160051314039977118584174167\dots$ | ||
| Stable Faltings height: | $0.10160051314039977118584174167\dots$ | ||
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: | $0.90398223907536685825088804389\dots$ | ||
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: | $ 3 $ = $ 1\cdot1\cdot3\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) $ ≈ $ 2.7119467172261005747526641317 $ | ||
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);
\( q + q^{2} + q^{3} + q^{4} - 3 q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - 3 q^{10} + q^{12} + 5 q^{13} + q^{14} - 3 q^{15} + q^{16} + q^{18} + 8 q^{19} + O(q^{20}) \)
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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: | 360 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $23$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
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 $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $3$ | 3B.1.2 | 3.8.0.2 |
$p$-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 | 7 | 23 |
|---|---|---|---|---|
| Reduction type | split | split | split | nonsplit |
| $\lambda$-invariant(s) | 1 | 1 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 966.i
consists of 2 curves linked by isogenies of
degree 3.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | Not in database |
| $3$ | 3.1.3864.1 | \(\Z/2\Z\) | Not in database |
| $3$ | 3.1.972.2 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.0.57691436544.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $6$ | 6.0.2834352.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $6$ | 6.0.44791488.1 | \(\Z/6\Z\) | Not in database |
| $9$ | 9.1.1471660546962235392.2 | \(\Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $18$ | 18.0.47963335115045709516582450861214474176867072.1 | \(\Z/9\Z\) | Not in database |
| $18$ | 18.0.6497354296455557525005155040852180992.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $18$ | 18.0.3470756510165292232129412919557456735036768256.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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.