Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2+85x-2437\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z+85xz^2-2437z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+1365x-154586\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(19, 66\right) \)
Integral points
\( \left(19, 66\right) \), \( \left(19, -86\right) \)
Invariants
Conductor: | \( 342 \) | = | $2 \cdot 3^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-2560108032 $ | = | $-1 \cdot 2^{9} \cdot 3^{6} \cdot 19^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{94196375}{3511808} \) | = | $2^{-9} \cdot 5^{3} \cdot 7^{3} \cdot 13^{3} \cdot 19^{-3}$ | 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: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.48516961095724358095196192008\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.064136533376811264745660698381\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.01874591611381\dots$ | |||
Szpiro ratio: | $4.987718092645647\dots$ |
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: | $0.69433745979450682760305816073\dots$ | 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: | $ 27 $ = $ 3^{2}\cdot1\cdot3 $ | 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) $ ≈ $ 2.0830123793835204828091744822 $ | 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
$\displaystyle 2.083012379 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.694337 \cdot 1.000000 \cdot 27}{3^2} \approx 2.083012379$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 180 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$3$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$19$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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 |
---|---|---|
$3$ | 3Cs.1.1 | 9.72.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4104 = 2^{3} \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 19 & 54 \\ 792 & 3547 \end{array}\right),\left(\begin{array}{rr} 1595 & 592 \\ 3192 & 1291 \end{array}\right),\left(\begin{array}{rr} 892 & 27 \\ 2373 & 622 \end{array}\right),\left(\begin{array}{rr} 1 & 27 \\ 27 & 730 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 2025 & 4078 \end{array}\right),\left(\begin{array}{rr} 3079 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4051 & 54 \\ 4050 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 43 & 30 \\ 3918 & 1111 \end{array}\right)$.
The torsion field $K:=\Q(E[4104])$ is a degree-$45954293760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4104\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 342.e
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 38.a3, its twist by $-3$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | 2.0.3.1-1444.2-b5 |
$3$ | 3.1.152.1 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.3511808.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.623808.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$9$ | 9.3.75006330133563.2 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$12$ | 12.0.8990607867641856.3 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.43564677551979246963.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.358752951832597688547.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.16877848680315122776257224907.3 | \(\Z/3\Z \oplus \Z/9\Z\) | 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 | split | add | ss | ord | ord | ord | ord | split | ord | ord | ord | ord | ss | ord | ss |
$\lambda$-invariant(s) | 1 | - | 0,0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 |
$\mu$-invariant(s) | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.