Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-40190455x-98072518885\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-40190455xz^2-98072518885z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-52086829707x-4575515180609466\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-14641/4, 14641/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 2730 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $49441793310$ | = | $2 \cdot 3^{8} \cdot 5 \cdot 7^{3} \cdot 13^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{7179471593960193209684686321}{49441793310} \) | = | $2^{-1} \cdot 3^{-8} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{6} \cdot 13^{-3} \cdot 107^{3} \cdot 109^{3} \cdot 1367^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5877817507268186953864463263$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.5877817507268186953864463263$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.087781017100558$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.106744453880887$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.059960994022966622286483262967$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 1\cdot2^{3}\cdot1\cdot3\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $4.3171915696535968046267949336 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
|
BSD formula
$$\begin{aligned} 4.317191570 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.059961 \cdot 1.000000 \cdot 72}{2^2} \\ & \approx 4.317191570\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 110592 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 | 8.12.0.6 |
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 2734 & 3 \\ 2775 & 34 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 9614 & 1691 \end{array}\right),\left(\begin{array}{rr} 8646 & 10489 \\ 5915 & 7736 \end{array}\right),\left(\begin{array}{rr} 10897 & 24 \\ 10896 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1821 & 3644 \\ 20 & 5541 \end{array}\right),\left(\begin{array}{rr} 1696 & 21 \\ 3915 & 10546 \end{array}\right),\left(\begin{array}{rr} 4696 & 3 \\ 1101 & 10834 \end{array}\right),\left(\begin{array}{rr} 8752 & 21 \\ 10635 & 10546 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$4869303828480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 455 = 5 \cdot 7 \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 10 = 2 \cdot 5 \) |
| $5$ | split multiplicative | $6$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 2730.bd
consists of 8 curves linked by isogenies of
degrees dividing 12.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{910}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-65}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | not in database |
| $3$ | 3.1.24300.4 | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-14}, \sqrt{-65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.45660160.5 | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{910})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-14})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-65})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.1771470000.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.284784729465600000.3 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.25924872960000.14 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.103784522400000.8 | \(\Z/12\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | deg 8 | \(\Z/24\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/24\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.0.19918452952908246993558395314624812290700000000.1 | \(\Z/18\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | split | split | split | split | split |
| $\lambda$-invariant(s) | 3 | 11 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 2 | 1 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.