Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-161195x-24887959\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-161195xz^2-24887959z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2579115x-1595408474\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8043/4, 697481/8)$ | $7.7283006252310354645082943206$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 80586 \) | = | $2 \cdot 3^{2} \cdot 11^{2} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-392500362202542$ | = | $-1 \cdot 2 \cdot 3^{7} \cdot 11^{6} \cdot 37^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{358667682625}{303918} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 5^{3} \cdot 7^{6} \cdot 29^{3} \cdot 37^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7276270804285360951588876218$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.020626700304704022569706785643$ |
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| $abc$ quality: | $Q$ | ≈ | $1.038169487521527$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.212253939456915$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.7283006252310354645082943206$ |
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| Real period: | $\Omega$ | ≈ | $0.11912689155823307724670887150$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot2^{2}\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.047781166135869418095988423 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.047781166 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.119127 \cdot 7.728301 \cdot 12}{1^2} \\ & \approx 11.047781166\end{aligned}$$
Modular invariants
Modular form 80586.2.a.ba
For more coefficients, see the Downloads section to the right.
| Modular degree: | 388800 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $11$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $37$ | $3$ | $I_{3}$ | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9768 = 2^{3} \cdot 3 \cdot 11 \cdot 37 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 4885 & 1782 \\ 5775 & 5347 \end{array}\right),\left(\begin{array}{rr} 3551 & 0 \\ 0 & 9767 \end{array}\right),\left(\begin{array}{rr} 7327 & 1782 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2254 & 3069 \\ 6215 & 3662 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 9763 & 6 \\ 9762 & 7 \end{array}\right),\left(\begin{array}{rr} 3961 & 1782 \\ 3003 & 5347 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9768])$ is a degree-$110834948505600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9768\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 40293 = 3^{2} \cdot 11^{2} \cdot 37 \) |
| $3$ | additive | $8$ | \( 242 = 2 \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 666 = 2 \cdot 3^{2} \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 80586.ba
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 222.e1, its twist by $33$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.888.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.700227072.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.3772522512.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.1049552064.1 | \(\Z/6\Z\) | 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 |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.818149109701156089099963023360754738073338624.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.2256970090595694635954288563231141310496768.1 | \(\Z/6\Z\) | 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 | split | add | ss | ord | add | ord | ord | ord | ord | ss | ord | split | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | 5,1 | 3 | - | 1 | 1 | 1 | 1 | 1,1 | 1 | 2 | 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.