Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2+x\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z+xz^2\)
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(dehomogenize, simplify) |
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\(y^2=x^3+864x-432\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(0, 0\right) \)
Integral points
\( \left(0, 0\right) \), \( \left(0, -1\right) \)
Invariants
| Conductor: | \( 19 \) | = | $19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-19 $ | = | $-1 \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{32768}{19} \) | = | $2^{15} \cdot 19^{-1}$ | 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: | $-1.0651731316767918878751503533\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-1.0651731316767918878751503533\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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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: | $4.0792792004649324322095526836\dots$ | 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: | $3$ | 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) $ ≈ $ 0.45325324449610360357883918707 $ | 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
$\displaystyle 0.453253244 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.079279 \cdot 1.000000 \cdot 1}{3^2} \approx 0.453253244$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 3 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There is only one prime of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $19$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $3$ | 3B.1.1 | 27.72.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1026 = 2 \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 31 & 36 \\ 334 & 421 \end{array}\right),\left(\begin{array}{rr} 994 & 9 \\ 689 & 790 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 973 & 54 \\ 972 & 55 \end{array}\right),\left(\begin{array}{rr} 149 & 681 \\ 271 & 448 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right)$.
The torsion field $K:=\Q(E[1026])$ is a degree-$179508960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1026\Z)$.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 19 |
|---|---|---|---|
| Reduction type | ss | ord | split |
| $\lambda$-invariant(s) | 0,3 | 0 | 1 |
| $\mu$-invariant(s) | 0,0 | 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 and 9.
Its isogeny class 19a
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.76.1 | \(\Z/6\Z\) | Not in database |
| $3$ | 3.3.361.1 | \(\Z/9\Z\) | 3.3.361.1-19.1-a3 |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.0.3518667.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $6$ | 6.0.9747.1 | \(\Z/9\Z\) | Not in database |
| $9$ | 9.3.57207791296.1 | \(\Z/18\Z\) | Not in database |
| $12$ | 12.2.937292452593664.2 | \(\Z/12\Z\) | Not in database |
| $18$ | 18.0.43564677551979246963.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
| $18$ | 18.0.64417171850299425397321728.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $18$ | 18.0.494296175215808851968.1 | \(\Z/18\Z\) | Not in database |
| $18$ | 18.0.62181896314367173832704.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
We only show fields where the torsion growth is primitive.