Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-33x-383\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-33xz^2-383z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2700x-287280\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(21, 88)$ | $1.6903056979163862357229433568$ | $\infty$ |
Integral points
\((21,\pm 88)\)
Invariants
| Conductor: | $N$ | = | \( 126400 \) | = | $2^{6} \cdot 5^{2} \cdot 79$ |
|
| Discriminant: | $\Delta$ | = | $-64716800$ | = | $-1 \cdot 2^{15} \cdot 5^{2} \cdot 79 $ |
|
| j-invariant: | $j$ | = | \( -\frac{5000}{79} \) | = | $-1 \cdot 2^{3} \cdot 5^{4} \cdot 79^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.18010942261036896989450809164$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.95456420516191272931049194905$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.772506327427777$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.1687047695995183$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6903056979163862357229433568$ |
|
| Real period: | $\Omega$ | ≈ | $0.84169923687689858037845482610$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.6909160640995830135896389155 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.690916064 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.841699 \cdot 1.690306 \cdot 4}{1^2} \\ & \approx 5.690916064\end{aligned}$$
Modular invariants
Modular form 126400.2.a.x
For more coefficients, see the Downloads section to the right.
| Modular degree: | 25344 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 6 | 15 | 0 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $79$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 632 = 2^{3} \cdot 79 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 631 & 0 \end{array}\right),\left(\begin{array}{rr} 161 & 2 \\ 161 & 3 \end{array}\right),\left(\begin{array}{rr} 631 & 2 \\ 630 & 3 \end{array}\right),\left(\begin{array}{rr} 317 & 2 \\ 317 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 159 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[632])$ is a degree-$29530275840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/632\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 1975 = 5^{2} \cdot 79 \) |
| $5$ | additive | $10$ | \( 5056 = 2^{6} \cdot 79 \) |
| $79$ | split multiplicative | $80$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 126400cd consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 63200b1, its twist by $-8$.
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 |
|---|---|---|---|
| $3$ | 3.1.15800.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.157772480000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | 79 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | add | ss | ord | ss | ord | ss | ord | ord | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | - | 1 | - | 3,1 | 1 | 3,1 | 1 | 3,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| $\mu$-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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.