Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-11x-14\)
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(homogenize, simplify) |
\(y^2z=x^3-11xz^2-14z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-11x-14\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-2, 0\right) \)
Integral points
\( \left(-2, 0\right) \)
Invariants
Conductor: | \( 32 \) | = | $2^{5}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $512 $ | = | $2^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( 287496 \) | = | $2^{3} \cdot 3^{3} \cdot 11^{3}$ | 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: | \(\Z[\sqrt{-4}]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $-0.61738574535156420883504296185\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.1372461307715231908979670529\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.172456969504371\dots$ | |||
Szpiro ratio: | $5.426636471615072\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $2.6220575542921198104648395899\dots$ | 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: | $ 1 $ | 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: | $2$ | 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) $ ≈ $ 0.65551438857302995261620989747 $ | 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
$\displaystyle 0.655514389 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.622058 \cdot 1.000000 \cdot 1}{2^2} \approx 0.655514389$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 4 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 2 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There is only one prime $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_0^{*}$ | additive | -1 | 5 | 9 | 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 |
---|---|---|
$2$ | 2B | 16.192.3.554 |
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | additive | $4$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 32a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.8.1-32.1-a5 |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | 2.0.4.1-64.1-CMa2 |
$2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | 2.0.8.1-32.1-a3 |
$4$ | \(\Q(\zeta_{8})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | 4.2.2048.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | 4.0.512.1 | \(\Z/8\Z\) | not in database |
$8$ | 8.0.16777216.2 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$8$ | \(\Q(\zeta_{16})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.4194304.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.2.143327232.1 | \(\Z/6\Z\) | not in database |
$8$ | 8.0.8192000.1 | \(\Z/20\Z\) | not in database |
$16$ | 16.4.73786976294838206464.5 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.0.18014398509481984.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.0.288230376151711744.2 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$16$ | 16.0.20542695432781824.1 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
$16$ | 16.4.1048576000000000000.1 | \(\Z/10\Z\) | not in database |
$16$ | 16.4.5258930030792146944.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$16$ | 16.0.17179869184000000.1 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
$16$ | 16.0.5258930030792146944.3 | \(\Z/12\Z\) | not in database |
$16$ | deg 16 | \(\Z/40\Z\) | 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 |
---|---|
Reduction type | add |
$\lambda$-invariant(s) | - |
$\mu$-invariant(s) | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.