Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-220865x+122211137\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-220865xz^2+122211137z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-17890092x+89038248624\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-641, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-641:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-5772, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-641, 0\right) \)
\([-641:0:1]\)
\( \left(-641, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 438080 \) | = | $2^{6} \cdot 5 \cdot 37^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-5754846337304166400$ | = | $-1 \cdot 2^{16} \cdot 5^{2} \cdot 37^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{7086244}{34225} \) | = | $-1 \cdot 2^{2} \cdot 5^{-2} \cdot 11^{6} \cdot 37^{-2}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2842528131852303210357017773$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.44540238388347564703798888683$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9379917665694112$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9079175502762338$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.20834734443060502197980391844$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $6.6671150217793607033537253900 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 6.667115022 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.208347 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 6.667115022\end{aligned}$$
Modular invariants
Modular form 438080.2.a.dn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8404992 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | 1 | 6 | 16 | 0 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $37$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|---|
| $2$ | 2B | 4.6.0.5 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1480 = 2^{3} \cdot 5 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1111 & 8 \\ 185 & 1 \end{array}\right),\left(\begin{array}{rr} 1473 & 8 \\ 1472 & 9 \end{array}\right),\left(\begin{array}{rr} 297 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 479 & 1478 \\ 0 & 1479 \end{array}\right),\left(\begin{array}{rr} 739 & 1472 \\ 0 & 1479 \end{array}\right)$.
The torsion field $K:=\Q(E[1480])$ is a degree-$27988623360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1480\Z)$.
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 | $2$ | \( 1369 = 37^{2} \) |
| $5$ | split multiplicative | $6$ | \( 87616 = 2^{6} \cdot 37^{2} \) |
| $37$ | additive | $722$ | \( 320 = 2^{6} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 438080dn
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1480c2, its twist by $296$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.