Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-53103x-4695698\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-53103xz^2-4695698z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-53103x-4695698\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(282749/121, 149588712/1331)$ | $10.815900206288482276769084328$ | $\infty$ |
| $(-127, 0)$ | $0$ | $2$ |
| $(266, 0)$ | $0$ | $2$ |
Integral points
\( \left(-139, 0\right) \), \( \left(-127, 0\right) \), \( \left(266, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 94320 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 131$ |
|
| Discriminant: | $\Delta$ | = | $58368377606400$ | = | $2^{8} \cdot 3^{12} \cdot 5^{2} \cdot 131^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{88738766769616}{312759225} \) | = | $2^{4} \cdot 3^{-6} \cdot 5^{-2} \cdot 31^{3} \cdot 131^{-2} \cdot 571^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5036613110558065855929369259$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.49225704634845486695049289313$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8966396359967733$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.863439519162789$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.815900206288482276769084328$ |
|
| Real period: | $\Omega$ | ≈ | $0.31456604073107995153857017611$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.8046298096692775321456592905 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.804629810 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.314566 \cdot 10.815900 \cdot 32}{4^2} \\ & \approx 6.804629810\end{aligned}$$
Modular invariants
Modular form 94320.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 282624 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $131$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7860 = 2^{2} \cdot 3 \cdot 5 \cdot 131 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4717 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6421 & 4 \\ 4982 & 9 \end{array}\right),\left(\begin{array}{rr} 5239 & 7858 \\ 0 & 7859 \end{array}\right),\left(\begin{array}{rr} 3927 & 7858 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 7857 & 4 \\ 7856 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[7860])$ is a degree-$13466179584000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7860\Z)$.
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 | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $6$ | \( 10480 = 2^{4} \cdot 5 \cdot 131 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 18864 = 2^{4} \cdot 3^{2} \cdot 131 \) |
| $131$ | split multiplicative | $132$ | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 94320c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 15720b2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-131})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{393})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | 131 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | nonsplit | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | - | - | 1 | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.