Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+3741x-6554\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+3741xz^2-6554z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+4848957x-320318658\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(14, 213)$ | $0.74876839924882064959256043723$ | $\infty$ |
| $(7/4, -11/8)$ | $0$ | $2$ |
Integral points
\( \left(14, 213\right) \), \( \left(14, -228\right) \), \( \left(58, 608\right) \), \( \left(58, -667\right) \), \( \left(308, 5358\right) \), \( \left(308, -5667\right) \)
Invariants
| Conductor: | $N$ | = | \( 2730 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $-3372408585000$ | = | $-1 \cdot 2^{3} \cdot 3^{2} \cdot 5^{4} \cdot 7^{8} \cdot 13 $ |
|
| j-invariant: | $j$ | = | \( \frac{5792335463322071}{3372408585000} \) | = | $2^{-3} \cdot 3^{-2} \cdot 5^{-4} \cdot 7^{-8} \cdot 13^{-1} \cdot 179591^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0933230466280552671581052838$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0933230466280552671581052838$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0195198747653818$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.587342138671309$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.74876839924882064959256043723$ |
|
| Real period: | $\Omega$ | ≈ | $0.46906544610545407490964832915$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2\cdot2\cdot2^{3}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.8097710657865184147160534421 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.809771066 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.469065 \cdot 0.748768 \cdot 32}{2^2} \\ & \approx 2.809771066\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6144 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $2$ | 2B | 8.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 520 = 2^{3} \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 328 & 73 \\ 353 & 400 \end{array}\right),\left(\begin{array}{rr} 513 & 8 \\ 512 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 164 & 1 \\ 423 & 6 \end{array}\right),\left(\begin{array}{rr} 417 & 8 \\ 108 & 33 \end{array}\right),\left(\begin{array}{rr} 456 & 203 \\ 329 & 358 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 514 & 515 \end{array}\right)$.
The torsion field $K:=\Q(E[520])$ is a degree-$402554880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/520\Z)$.
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 | $4$ | \( 13 \) |
| $3$ | split multiplicative | $4$ | \( 455 = 5 \cdot 7 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 2730m
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-26}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-13}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1639853447970816.38 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.443023360000.10 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1000887114240000.19 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.7592405385166875.10 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | split | ss | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | 2 | 1 | 2 | 3,1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 2 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.