Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2+7x\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z+7xz^2\) | (dehomogenize, simplify) |
\(y^2=x^3+540x+1647\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(0, 0\right) \)
Integral points
\( \left(0, 0\right) \)
Invariants
Conductor: | \( 336 \) | = | $2^{4} \cdot 3 \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-21168 $ | = | $-1 \cdot 2^{4} \cdot 3^{3} \cdot 7^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{2048000}{1323} \) | = | $2^{14} \cdot 3^{-3} \cdot 5^{3} \cdot 7^{-2}$ | 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: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $-0.48107077216270159681646974074\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.71211983234935003328888044789\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1084282779180015\dots$ | |||
Szpiro ratio: | $2.9748379412936607\dots$ |
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: | $2.3898781773721729422182513156\dots$ | 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 $ = $ 1\cdot1\cdot2 $ | 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) $ ≈ $ 1.1949390886860864711091256578 $ | 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
$\displaystyle 1.194939089 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.389878 \cdot 1.000000 \cdot 2}{2^2} \approx 1.194939089$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 24 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II$ | Additive | -1 | 4 | 4 | 0 |
$3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 84 = 2^{2} \cdot 3 \cdot 7 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 73 & 12 \\ 72 & 13 \end{array}\right),\left(\begin{array}{rr} 45 & 74 \\ 70 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 73 & 12 \\ 18 & 73 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 34 & 75 \end{array}\right),\left(\begin{array}{rr} 62 & 11 \\ 45 & 64 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[84])$ is a degree-$96768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/84\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 336.b
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 84.b4, its twist by $-4$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.3.1-37632.2-a5 |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/6\Z\) | 2.0.4.1-1764.1-a2 |
$4$ | 4.2.9408.2 | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\zeta_{12})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.2.16595712.3 | \(\Z/6\Z\) | Not in database |
$8$ | 8.0.146313216.2 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.796594176.14 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.88510464.1 | \(\Z/12\Z\) | Not in database |
$12$ | 12.0.275417656786944.1 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | 16.0.21407557176262656.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | 16.0.634562281237118976.3 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$18$ | 18.0.19913236773125052849281040384.1 | \(\Z/18\Z\) | 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 | 7 |
---|---|---|---|
Reduction type | add | nonsplit | nonsplit |
$\lambda$-invariant(s) | - | 0 | 0 |
$\mu$-invariant(s) | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.