Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-48344651x-129214090302\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-48344651xz^2-129214090302z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-62654667075x-6028424633117250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-4133, 2066)$ | $0$ | $2$ |
Integral points
\( \left(-4133, 2066\right) \)
Invariants
| Conductor: | $N$ | = | \( 414050 \) | = | $2 \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $19104008934365554062500$ | = | $2^{2} \cdot 5^{7} \cdot 7^{8} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{1408317602329}{2153060} \) | = | $2^{-2} \cdot 5^{-1} \cdot 7^{-2} \cdot 11^{3} \cdot 13^{-3} \cdot 1019^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1749869209126533773675649557$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.11483821143717816948776519658$ |
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| $abc$ quality: | $Q$ | ≈ | $0.895954553441358$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.002048234719847$ | |||
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: | $\Omega$ | ≈ | $0.057260129306088526951206260085$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.91616206889741643121930016136 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.916162069 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.057260 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 0.916162069\end{aligned}$$
Modular invariants
Modular form 414050.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 55738368 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
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 | 2.3.0.1 | $3$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1670 & 10917 \\ 4227 & 8 \end{array}\right),\left(\begin{array}{rr} 3119 & 10908 \\ 7794 & 10847 \end{array}\right),\left(\begin{array}{rr} 5461 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 7289 & 2 \\ 1878 & 13 \end{array}\right),\left(\begin{array}{rr} 10909 & 12 \\ 10908 & 13 \end{array}\right),\left(\begin{array}{rr} 5919 & 7738 \\ 3206 & 5021 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 10870 & 10911 \end{array}\right),\left(\begin{array}{rr} 10910 & 10917 \\ 6579 & 8 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$19477215313920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 207025 = 5^{2} \cdot 7^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 16562 = 2 \cdot 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 414050n
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 910c1, its twist by $-455$.
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 $0$.