Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-7017x-235305\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-7017xz^2-235305z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-9094059x-10841970906\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 41574 \) | = | $2 \cdot 3 \cdot 13^{2} \cdot 41$ |
|
| Discriminant: | $\Delta$ | = | $-1215892494336$ | = | $-1 \cdot 2^{11} \cdot 3 \cdot 13^{6} \cdot 41 $ |
|
| j-invariant: | $j$ | = | \( -\frac{7916293657}{251904} \) | = | $-1 \cdot 2^{-11} \cdot 3^{-1} \cdot 41^{-1} \cdot 1993^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0948844004518413195561467266$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.18759027827892704847059699418$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.931505436187378$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5951655275588843$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.26032651311599483577233596215$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 11 $ = $ 11\cdot1\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.8635916442759431934956955836 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.863591644 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.260327 \cdot 1.000000 \cdot 11}{1^2} \\ & \approx 2.863591644\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 102960 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $11$ | $I_{11}$ | split multiplicative | -1 | 1 | 11 | 11 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $41$ | $1$ | $I_{1}$ | nonsplit 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 \( 984 = 2^{3} \cdot 3 \cdot 41 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 247 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 493 & 2 \\ 493 & 3 \end{array}\right),\left(\begin{array}{rr} 457 & 2 \\ 457 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 983 & 0 \end{array}\right),\left(\begin{array}{rr} 983 & 2 \\ 982 & 3 \end{array}\right),\left(\begin{array}{rr} 329 & 2 \\ 329 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[984])$ is a degree-$101567692800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/984\Z)$.
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 | $4$ | \( 20787 = 3 \cdot 13^{2} \cdot 41 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 13858 = 2 \cdot 13^{2} \cdot 41 \) |
| $11$ | good | $2$ | \( 20787 = 3 \cdot 13^{2} \cdot 41 \) |
| $13$ | additive | $86$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
| $41$ | nonsplit multiplicative | $42$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 41574m consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 246g1, 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 |
|---|---|---|---|
| $3$ | 3.1.984.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.952763904.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | nonsplit | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | nonsplit | ord | ord |
| $\lambda$-invariant(s) | 1 | 0 | 0 | 0 | 2 | - | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.