Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2-2678049x+1685955405\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z-2678049xz^2+1685955405z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3470751936x+78701584408848\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(23041/25, 151361/125)$ | $3.6152310549732122407420087763$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 66139 \) | = | $19 \cdot 59^{2}$ |
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| Discriminant: | $\Delta$ | = | $-801430139179$ | = | $-1 \cdot 19 \cdot 59^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{50357871050752}{19} \) | = | $-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0722078789441775288281200704$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.033439156991317803520094883539$ |
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| $abc$ quality: | $Q$ | ≈ | $1.104947099482495$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.046653518794261$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.6152310549732122407420087763$ |
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| Real period: | $\Omega$ | ≈ | $0.53730166399491128500717851282$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8849393231263710638077877659 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.884939323 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.537302 \cdot 3.615231 \cdot 2}{1^2} \\ & \approx 3.884939323\end{aligned}$$
Modular invariants
Modular form 66139.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 620136 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $59$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 60534 = 2 \cdot 3^{3} \cdot 19 \cdot 59 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 27613 & 7257 \\ 43719 & 14456 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 54712 & 53773 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 57526 & 16461 \\ 33217 & 18940 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 42065 & 0 \\ 0 & 60533 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 60481 & 54 \\ 60480 & 55 \end{array}\right)$.
The torsion field $K:=\Q(E[60534])$ is a degree-$2137693220697600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/60534\Z)$.
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 |
|---|---|---|---|
| $19$ | split multiplicative | $20$ | \( 3481 = 59^{2} \) |
| $59$ | additive | $1742$ | \( 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 66139a
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19a2, its twist by $-59$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{177}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.240886769931.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.526819365839097.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.2.32029265808.2 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.2129654040585234330084129.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.20668358903285946754787569386475185156096.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.216198334254740720361329607016400447927586848575488.1 | \(\Z/18\Z\) | 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 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ord | ord | ord | ord | ord | ord | split | ss | ord | ord | ord | ord | ord | ord | add |
| $\lambda$-invariant(s) | 2,5 | 5 | 3 | 1 | 1 | 1 | 1 | 4 | 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 | 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.