Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2+2x+1\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z+2xz^2+z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+135x+297\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(0, 1\right) \)
Integral points
\((0,\pm 1)\)
Invariants
Conductor: | \( 92 \) | = | $2^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-368 $ | = | $-1 \cdot 2^{4} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{32000}{23} \) | = | $2^{8} \cdot 5^{3} \cdot 23^{-1}$ | 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: | $-0.81926932071563170653554343467\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.0503183809022801430079541418\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.7198241144476085\dots$ | |||
Szpiro ratio: | $2.907274340677134\dots$ |
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: | $3.4103924343412202787913767145\dots$ | 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: | $ 3 $ = $ 3\cdot1 $ | 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) $ ≈ $ 1.1367974781137400929304589048 $ | 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
$\displaystyle 1.136797478 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.410392 \cdot 1.000000 \cdot 3}{3^2} \approx 1.136797478$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $3$ | $IV$ | Additive | -1 | 2 | 4 | 0 |
$23$ | $1$ | $I_{1}$ | Non-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 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 138 = 2 \cdot 3 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 97 & 6 \\ 15 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 133 & 6 \\ 132 & 7 \end{array}\right),\left(\begin{array}{rr} 24 & 121 \\ 115 & 93 \end{array}\right)$.
The torsion field $K:=\Q(E[138])$ is a degree-$4809024$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/138\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 92a
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.23.1 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.12167.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.120891312.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$9$ | 9.3.352519065792.2 | \(\Z/9\Z\) | Not in database |
$12$ | 12.2.892554561978368.10 | \(\Z/12\Z\) | Not in database |
$18$ | 18.0.934632642051670886361280512.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
$p$ | 2 | 3 | 23 |
---|---|---|---|
Reduction type | add | ord | nonsplit |
$\lambda$-invariant(s) | - | 2 | 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 $0$.