Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-17968563x+29315410367\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-17968563xz^2+29315410367z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-23287257675x+1367809647855750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(478447/196, -1885029/2744)$ | $3.1607323063663754346948774355$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 6050 \) | = | $2 \cdot 5^{2} \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-44579948532968750$ | = | $-1 \cdot 2 \cdot 5^{7} \cdot 11^{11} $ |
|
| j-invariant: | $j$ | = | \( -\frac{23178622194826561}{1610510} \) | = | $-1 \cdot 2^{-1} \cdot 5^{-1} \cdot 11^{-5} \cdot 285121^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6487421733790569495196387461$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.64507558076282149018828729051$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0129363433875729$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.0885769041134905$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1607323063663754346948774355$ |
|
| Real period: | $\Omega$ | ≈ | $0.27297354228957140782210880320$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.9023703511833308465102610286 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.902370351 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.272974 \cdot 3.160732 \cdot 8}{1^2} \\ & \approx 6.902370351\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 288000 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $11$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 440 = 2^{3} \cdot 5 \cdot 11 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 103 & 210 \\ 265 & 149 \end{array}\right),\left(\begin{array}{rr} 431 & 10 \\ 430 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 430 \\ 155 & 389 \end{array}\right),\left(\begin{array}{rr} 111 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 221 & 10 \\ 225 & 51 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 385 & 321 \end{array}\right)$.
The torsion field $K:=\Q(E[440])$ is a degree-$202752000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/440\Z)$.
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 | $4$ | \( 3025 = 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 242 = 2 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 6050z
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 110a2, its twist by $-55$.
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 |
|---|---|---|---|
| $3$ | 3.1.440.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.4.15125.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.85184000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.786381835937500000000.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | 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 | ord | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | - | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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.