Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+y=x^3+x^2-1577x-26178\)
|
(homogenize, simplify) |
|
\(y^2z+yz^2=x^3+x^2z-1577xz^2-26178z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2044224x-1196820144\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(56, 253\right) \) | $1.8117456943965924069931348530$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([56:253:1]\) | $1.8117456943965924069931348530$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2028, 54756\right) \) | $1.8117456943965924069931348530$ | $\infty$ |
Integral points
\( \left(56, 253\right) \), \( \left(56, -254\right) \)
\([56:253:1]\), \([56:-254:1]\)
\((2028,\pm 54756)\)
Invariants
| Conductor: | $N$ | = | \( 3211 \) | = | $13^{2} \cdot 19$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-33107082931$ | = | $-1 \cdot 13^{6} \cdot 19^{3} $ |
|
| j-invariant: | $j$ | = | \( -\frac{89915392}{6859} \) | = | $-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.76660769138803132584921598591$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.51586698734273704217752773487$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0331037033479094$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.189557234498756$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8117456943965924069931348530$ |
|
| Real period: | $\Omega$ | ≈ | $0.37712949549256563067696589366$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $1.3665254793772297695137809350 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 1.366525479 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.377129 \cdot 1.811746 \cdot 2}{1^2} \\ & \approx 1.366525479\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2160 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{3}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3Cs | 9.36.0.2 | $36$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13338 = 2 \cdot 3^{3} \cdot 13 \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 12637 & 2106 \\ 4875 & 4915 \end{array}\right),\left(\begin{array}{rr} 13285 & 54 \\ 13284 & 55 \end{array}\right),\left(\begin{array}{rr} 19 & 54 \\ 5922 & 5599 \end{array}\right),\left(\begin{array}{rr} 1 & 27 \\ 27 & 730 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1025 & 0 \\ 0 & 13337 \end{array}\right),\left(\begin{array}{rr} 43 & 30 \\ 9048 & 10345 \end{array}\right),\left(\begin{array}{rr} 12221 & 11050 \\ 13026 & 6709 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[13338])$ is a degree-$4704570823680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13338\Z)$.
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 |
|---|---|---|---|
| $3$ | good | $2$ | \( 169 = 13^{2} \) |
| $13$ | additive | $86$ | \( 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 169 = 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 3211a
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19a1, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/3\Z\) | 2.2.13.1-361.1-a2 |
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/3\Z\) | 2.0.39.1-361.1-a2 |
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{13})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.12689872.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.342626544.5 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.17110429473515225901042431637.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.178981135747990596923237760813001483311.2 | \(\Z/9\Z\) | 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 | ord | ord | ord | ord | add | ord | nonsplit | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2,5 | 5 | 1 | 1 | 1 | - | 1 | 1 | 3,1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-invariant(s) | 0,0 | 1 | 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.