Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-1317939x-582908538\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-1317939xz^2-582908538z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1708048971x-27170560006074\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2653/4, 2649/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 35301 \) | = | $3 \cdot 7 \cdot 41^{2}$ |
|
| Discriminant: | $\Delta$ | = | $698265323427$ | = | $3 \cdot 7^{2} \cdot 41^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{53297461115137}{147} \) | = | $3^{-1} \cdot 7^{-2} \cdot 37633^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9309914167430936726916475412$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.074205383390939770758265854681$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0508747826837916$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.14610944172793$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.14090477981084871184044197945$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.56361911924339484736176791779 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
|
BSD formula
$$\begin{aligned} 0.563619119 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.140905 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 0.563619119\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 281600 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $41$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 | 16.24.0.13 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13776 = 2^{4} \cdot 3 \cdot 7 \cdot 41 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6397 & 13120 \\ 164 & 9513 \end{array}\right),\left(\begin{array}{rr} 3158 & 11849 \\ 13653 & 8570 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 13678 & 13763 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 13772 & 13773 \end{array}\right),\left(\begin{array}{rr} 8399 & 0 \\ 0 & 13775 \end{array}\right),\left(\begin{array}{rr} 13761 & 16 \\ 13760 & 17 \end{array}\right),\left(\begin{array}{rr} 6848 & 8405 \\ 7011 & 2338 \end{array}\right),\left(\begin{array}{rr} 2461 & 13120 \\ 8528 & 6069 \end{array}\right)$.
The torsion field $K:=\Q(E[13776])$ is a degree-$34126744780800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13776\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 5043 = 3 \cdot 41^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 11767 = 7 \cdot 41^{2} \) |
| $7$ | split multiplicative | $8$ | \( 5043 = 3 \cdot 41^{2} \) |
| $41$ | additive | $842$ | \( 21 = 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 35301e
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 21a5, its twist by $41$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-123}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-41}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-41})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-41}, \sqrt{42})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{14}, \sqrt{-41})\) | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.527354820864.5 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | 7 | 41 |
|---|---|---|---|---|
| Reduction type | ord | nonsplit | split | add |
| $\lambda$-invariant(s) | 5 | 0 | 1 | - |
| $\mu$-invariant(s) | 2 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.