Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-8765105x-9987590203\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-8765105xz^2-9987590203z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-140241675x-639346014650\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-14334664934833582500$ | = | $-1 \cdot 2^{2} \cdot 3^{9} \cdot 5^{4} \cdot 7^{6} \cdot 19^{5} $ |
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| j-invariant: | $j$ | = | \( -\frac{1389310279182025}{267418692} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-3} \cdot 5^{2} \cdot 19^{-5} \cdot 31^{3} \cdot 1231^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6762075228617928760820148817$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.61746699985538125296479611377$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0156937260858383$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.601813068614449$ | |||
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: | $\Omega$ | ≈ | $0.043870771017224477666516124238$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 120 $ = $ 2\cdot2\cdot3\cdot2\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.2644925220669373199819349086 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 5.264492522 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043871 \cdot 1.000000 \cdot 120}{1^2} \\ & \approx 5.264492522\end{aligned}$$
Modular invariants
Modular form 418950.2.a.ov
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20736000 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7980 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 4201 & 3430 \\ 2765 & 1191 \end{array}\right),\left(\begin{array}{rr} 3991 & 3430 \\ 1715 & 1191 \end{array}\right),\left(\begin{array}{rr} 7971 & 10 \\ 7970 & 11 \end{array}\right),\left(\begin{array}{rr} 6273 & 3430 \\ 6860 & 2689 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2659 & 4550 \\ 7595 & 6789 \end{array}\right),\left(\begin{array}{rr} 2279 & 0 \\ 0 & 7979 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 7925 & 7861 \end{array}\right)$.
The torsion field $K:=\Q(E[7980])$ is a degree-$11437513113600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7980\Z)$.
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 | $4$ | \( 209475 = 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19 \) |
| $3$ | additive | $6$ | \( 46550 = 2 \cdot 5^{2} \cdot 7^{2} \cdot 19 \) |
| $5$ | additive | $14$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 22050 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 418950.ov
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 2850.d1, its twist by $21$.
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 $0$.