Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-13x+21\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-13xz^2+21z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1080x+12096\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(3, 0\right) \)
Integral points
\( \left(3, 0\right) \)
Invariants
Conductor: | \( 256 \) | = | $2^{8}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $32768 $ | = | $2^{15} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( 8000 \) | = | $2^{6} \cdot 5^{3}$ | 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: | \(\Z[\sqrt{-2}]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $-0.40397265035989411975627760561\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.2704066260598257565278177574\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9029767420170889\dots$ | |||
Szpiro ratio: | $3.495723035582761\dots$ |
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: | $3.5623007922266199348181877138\dots$ | 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: | $ 2 $ = $ 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: | $2$ | 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) $ ≈ $ 1.7811503961133099674090938569 $ | 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
$\displaystyle 1.781150396 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.562301 \cdot 1.000000 \cdot 2}{2^2} \approx 1.781150396$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 16 | 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 is only one prime $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $III^{*}$ | additive | -1 | 8 | 15 | 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 | 16.192.5.602 |
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.
Its isogeny class 256.d
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 256.a2, its twist by $8$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.8.1-1024.1-d4 |
$4$ | \(\Q(\zeta_{16})^+\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | 4.0.6144.2 | \(\Z/6\Z\) | not in database |
$4$ | 4.2.18432.1 | \(\Z/6\Z\) | not in database |
$8$ | 8.0.67108864.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.4.67108864.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.339738624.9 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
$8$ | 8.0.150994944.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$8$ | 8.4.1358954496.3 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$12$ | 12.0.169075682574336.3 | \(\Z/18\Z\) | not in database |
$16$ | 16.0.18014398509481984.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.0.1846757322198614016.7 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
$16$ | 16.0.364791569817010176.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
$16$ | 16.8.29548117155177824256.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
$20$ | 20.0.84954018740373771557797888.1 | \(\Z/22\Z\) | not in database |
We only show fields where the torsion growth is primitive.
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 $0$.