Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-2065467x+1142549354\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-2065467xz^2+1142549354z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2065467x+1142549354\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(679, 7290)$ | $1.6014117569504721854208105015$ | $\infty$ |
| $(829, 0)$ | $0$ | $2$ |
Integral points
\((679,\pm 7290)\), \( \left(829, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 65520 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $1390572091883520$ | = | $2^{14} \cdot 3^{15} \cdot 5 \cdot 7 \cdot 13^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{326355561310674169}{465699780} \) | = | $2^{-2} \cdot 3^{-9} \cdot 5^{-1} \cdot 7^{-1} \cdot 13^{-2} \cdot 29^{3} \cdot 23741^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1771029331806027951618779509$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.93464960828660264004702321098$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9713151603119757$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.980671984386753$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6014117569504721854208105015$ |
|
| Real period: | $\Omega$ | ≈ | $0.40823231877418168853440521106$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.2299842788170202358690268269 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.229984279 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.408232 \cdot 1.601412 \cdot 32}{2^2} \\ & \approx 5.229984279\end{aligned}$$
Modular invariants
Modular form 65520.2.a.ca
For more coefficients, see the Downloads section to the right.
| Modular degree: | 995328 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $4$ | $I_{9}^{*}$ | additive | -1 | 2 | 15 | 9 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 32760 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 26236 & 27 \\ 13413 & 298 \end{array}\right),\left(\begin{array}{rr} 7252 & 32733 \\ 21531 & 32462 \end{array}\right),\left(\begin{array}{rr} 17665 & 36 \\ 11874 & 9841 \end{array}\right),\left(\begin{array}{rr} 32725 & 36 \\ 32724 & 37 \end{array}\right),\left(\begin{array}{rr} 9364 & 9 \\ 4759 & 178 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 16381 & 36 \\ 10 & 361 \end{array}\right),\left(\begin{array}{rr} 24569 & 32724 \\ 0 & 32759 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[32760])$ is a degree-$175294937825280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32760\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| $3$ | additive | $2$ | \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 13104 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 65520eb
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 2730p1, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.4542720.7 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{35})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.98747943840000.2 | \(\Z/6\Z\) | not in database |
| $6$ | 6.6.118497532608.2 | \(\Z/18\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.185726744985600.6 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | add | add | split | nonsplit | ord | split | ord | ord | ss | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 2 | 1 | 1 | 2 | 1 | 1 | 1,1 | 1,3 | 1 | 1 | 1 | 3 | 1,1 |
| $\mu$-invariant(s) | - | - | 0 | 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.