Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-36x\)
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(homogenize, simplify) |
\(y^2z=x^3-36xz^2\)
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(dehomogenize, simplify) |
\(y^2=x^3-36x\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(-3, 9\right)\)
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$\hat{h}(P)$ | ≈ | $0.88862587483961923979418733498$ |
Torsion generators
\( \left(0, 0\right) \), \( \left(6, 0\right) \)
Integral points
\( \left(-6, 0\right) \), \((-3,\pm 9)\), \((-2,\pm 8)\), \( \left(0, 0\right) \), \( \left(6, 0\right) \), \((12,\pm 36)\), \((18,\pm 72)\), \((294,\pm 5040)\)
Invariants
Conductor: | \( 576 \) | = | $2^{6} \cdot 3^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $2985984 $ | = | $2^{12} \cdot 3^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( 1728 \) | = | $2^{6} \cdot 3^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z[\sqrt{-1}]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $-0.068079601017509363137420343386\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.3105329259115095182522750833\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.88862587483961923979418733498\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $2.1409010280752311986343110517\dots$ | comment: Real Period
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);
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Tamagawa product: | $ 16 $ = $ 2^{2}\cdot2^{2} $ | comment: Tamagawa numbers
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)
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Torsion order: | $4$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 1.9024600490183925553056020004 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 1.902460049 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.140901 \cdot 0.888626 \cdot 16}{4^2} \approx 1.902460049$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 64 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 6 | 12 | 0 |
$3$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
Galois representations
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 |
---|---|---|
$2$ | 2Cs | 16.192.9.143 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 576.c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 32.a3, its twist by $24$.
The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $36$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | \(\Q(\zeta_{24})\) | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.2.573308928.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.2654208000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$16$ | 16.8.118192468620711297024.2 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | 16.0.115422332637413376.2 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | 16.0.29548117155177824256.4 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | 16.0.328683126924509184.1 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$16$ | 16.4.5258930030792146944.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | ord | ss | ss | ord | ord | ss | ss | ord | ss | ord | ord | ss | ss |
$\lambda$-invariant(s) | - | - | 3 | 1,1 | 1,3 | 1 | 1 | 1,3 | 1,1 | 1 | 1,1 | 1 | 1 | 1,1 | 1,1 |
$\mu$-invariant(s) | - | - | 0 | 0,0 | 0,0 | 0 | 0 | 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.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.