Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-3834x-91375\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3-3834xz^2-91375z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-61344x-5847984\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-\frac{143}{4}, -\frac{3}{8}\right)\) |
$\hat{h}(P)$ | ≈ | $1.8632435522123629777793621451$ |
Integral points
None
Invariants
Conductor: | \( 189 \) | = | $3^{3} \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $1240029 $ | = | $3^{11} \cdot 7 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{35184082944}{7} \) | = | $2^{15} \cdot 3 \cdot 7^{-1} \cdot 71^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.55863419056949930850003858972\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $-0.44842707404293457527893621079\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.086380796976415\dots$ | |||
Szpiro ratio: | $6.938258313316677\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.8632435522123629777793621451\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.60671751526442999639570955648\dots$ | 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);
|
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)
|
Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 1.1304624983307551039521390320 $ | 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
$\displaystyle 1.130462498 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.606718 \cdot 1.863244 \cdot 1}{1^2} \approx 1.130462498$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 108 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $1$ | $II^{*}$ | Additive | -1 | 3 | 11 | 0 |
$7$ | $1$ | $I_{1}$ | 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.1.2 | 9.24.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 126 = 2 \cdot 3^{2} \cdot 7 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 10 & 55 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 109 & 18 \\ 108 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 124 & 117 \\ 45 & 62 \end{array}\right),\left(\begin{array}{rr} 40 & 9 \\ 99 & 106 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[126])$ is a degree-$326592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/126\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 189.b
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-3969.2-a5 |
$3$ | 3.3.756.1 | \(\Z/2\Z\) | Not in database |
$3$ | 3.1.11907.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.12002256.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.425329947.3 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$6$ | 6.0.64827.1 | \(\Z/9\Z\) | Not in database |
$6$ | 6.0.8680203.6 | \(\Z/9\Z\) | Not in database |
$6$ | 6.0.1714608.1 | \(\Z/6\Z\) | Not in database |
$9$ | 9.3.20419675633441728.13 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.144054149089536.2 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.76944553869448769155761123.2 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.1250889458924921506935668314877952.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.29058879047370913731981312.3 | \(\Z/18\Z\) | Not in database |
$18$ | 18.0.520986863358984384396363313152.1 | \(\Z/18\Z\) | Not in database |
$18$ | 18.6.8756226212474450548549678204145664.2 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ss | add | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4,1 | - | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Additional information
This is the elliptic curve $E$ of smallest conductor such that $E(\mathbb{Q})$ has positive rank, but the specialization map $E(\mathbb{Q}) \to E(\mathbb{F}_p)$ fails to be surjective for every prime $p$ of good reduction. See this mathoverflow post.