Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-135542x+18845192\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-135542xz^2+18845192z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-175661811x+879768275022\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(885, 23893)$ | $0.27920969270057031077668176883$ | $\infty$ |
Integral points
\( \left(270, 1261\right) \), \( \left(270, -1532\right) \), \( \left(300, 2131\right) \), \( \left(300, -2432\right) \), \( \left(885, 23893\right) \), \( \left(885, -24779\right) \)
Invariants
| Conductor: | $N$ | = | \( 41574 \) | = | $2 \cdot 3 \cdot 13^{2} \cdot 41$ |
|
| Discriminant: | $\Delta$ | = | $5761506584411136$ | = | $2^{10} \cdot 3^{7} \cdot 13^{7} \cdot 41 $ |
|
| j-invariant: | $j$ | = | \( \frac{57053285789473}{1193647104} \) | = | $2^{-10} \cdot 3^{-7} \cdot 13^{-1} \cdot 41^{-1} \cdot 137^{3} \cdot 281^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8139787872215454872046466036$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.53150410849077711917790288282$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9195098289495989$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.4253581558771815$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.27920969270057031077668176883$ |
|
| Real period: | $\Omega$ | ≈ | $0.42660312685256020910342167287$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 56 $ = $ 2\cdot7\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.6702567654019220309469587037 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.670256765 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.426603 \cdot 0.279210 \cdot 56}{1^2} \\ & \approx 6.670256765\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 376320 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $41$ | $1$ | $I_{1}$ | 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 \( 6396 = 2^{2} \cdot 3 \cdot 13 \cdot 41 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 4265 & 2 \\ 4265 & 3 \end{array}\right),\left(\begin{array}{rr} 3199 & 2 \\ 3199 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3901 & 2 \\ 3901 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 6395 & 0 \end{array}\right),\left(\begin{array}{rr} 2953 & 2 \\ 2953 & 3 \end{array}\right),\left(\begin{array}{rr} 6395 & 2 \\ 6394 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[6396])$ is a degree-$166367880806400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6396\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 20787 = 3 \cdot 13^{2} \cdot 41 \) |
| $3$ | split multiplicative | $4$ | \( 13858 = 2 \cdot 13^{2} \cdot 41 \) |
| $5$ | good | $2$ | \( 20787 = 3 \cdot 13^{2} \cdot 41 \) |
| $7$ | good | $2$ | \( 13858 = 2 \cdot 13^{2} \cdot 41 \) |
| $13$ | additive | $98$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 41574.j consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 3198.f1, 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 |
|---|---|---|---|
| $3$ | 3.3.6396.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.261652787136.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 | nonsplit | split | ord | ss | ord | add | ord | ord | ord | ord | ord | ord | split | ord | ord |
| $\lambda$-invariant(s) | 3 | 2 | 1 | 1,1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.