Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-54x-88\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-54xz^2-88z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-864x-5616\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-6, 4)$ | $0.62108118407078765925978738169$ | $\infty$ |
| $(12, 31)$ | $0$ | $3$ |
Integral points
\( \left(-6, 4\right) \), \( \left(-6, -5\right) \), \( \left(-2, 3\right) \), \( \left(-2, -4\right) \), \( \left(12, 31\right) \), \( \left(12, -32\right) \), \( \left(40, 248\right) \), \( \left(40, -249\right) \), \( \left(48, 328\right) \), \( \left(48, -329\right) \)
Invariants
| Conductor: | $N$ | = | \( 189 \) | = | $3^{3} \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $\Delta$ | = | $6751269$ | = | $3^{9} \cdot 7^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | $j$ | = | \( \frac{884736}{343} \) | = | $2^{15} \cdot 3^{3} \cdot 7^{-3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.0093280462354444628024159712558$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.81463117026563780574401795644$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $Q$ | ≈ | $1.198754152359422$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.498606114463192$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Mordell-Weil rank: | $r$ | = | $ 1$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.62108118407078765925978738169$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $\Omega$ | ≈ | $1.8201525457932899891871286694$ | 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: | $\prod_{p}c_p$ | = | $ 9 $ = $ 3\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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ | 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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| 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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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 1.130462498 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.820153 \cdot 0.621081 \cdot 9}{3^2} \approx 1.130462498$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 36 | 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 at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $3$ | $IV^{*}$ | additive | -1 | 3 | 9 | 0 |
| $7$ | $3$ | $I_{3}$ | 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 | 3.24.0.1 |
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
$\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 34 & 9 \\ 27 & 76 \end{array}\right),\left(\begin{array}{rr} 109 & 18 \\ 108 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 116 & 117 \\ 9 & 106 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right)$.
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 $\GL_2(\Z/126\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $3$ | additive | $2$ | \( 1 \) |
| $7$ | split multiplicative | $8$ | \( 27 = 3^{3} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | 2.0.3.1-3969.2-a3 |
| $3$ | 3.3.756.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.6.12002256.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.1714608.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.3.136738899331083.8 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.144054149089536.2 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.476779061197529538836283.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.144784752906623254803.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.56092579770828152714549858667.2 | \(\Z/3\Z \oplus \Z/9\Z\) | 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 | 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.