Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-5990639055x+178464471262629\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-5990639055xz^2+178464471262629z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7763868215955x+8326554829252454574\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(219130352532872654/4909844977489, -22803457284577736023599/10879317968484743513)$ | $33.975497961697286143781028902$ | $\infty$ |
| $(44686, -22343)$ | $0$ | $2$ |
Integral points
\( \left(44686, -22343\right) \)
Invariants
| Conductor: | $N$ | = | \( 405042 \) | = | $2 \cdot 3 \cdot 11 \cdot 17 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $38373289533960741888$ | = | $2^{10} \cdot 3 \cdot 11 \cdot 17^{6} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{505384091400037554067434625}{815656731648} \) | = | $2^{-10} \cdot 3^{-1} \cdot 5^{3} \cdot 11^{-1} \cdot 17^{-6} \cdot 31^{6} \cdot 97^{3} \cdot 1709^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9131643742422945268143428260$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4409448846590742968098291101$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1764898982289331$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.130386591541099$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $33.975497961697286143781028902$ |
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| Real period: | $\Omega$ | ≈ | $0.093024498201658953761809064246$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.3211072980767532879598743435 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.321107298 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.093024 \cdot 33.975498 \cdot 8}{2^2} \\ & \approx 6.321107298\end{aligned}$$
Modular invariants
Modular form 405042.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 209018880 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $19$ | $2$ | $I_0^{*}$ | 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 | 8.6.0.4 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5016 = 2^{3} \cdot 3 \cdot 11 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2639 & 0 \\ 0 & 5015 \end{array}\right),\left(\begin{array}{rr} 2870 & 2907 \\ 589 & 1312 \end{array}\right),\left(\begin{array}{rr} 1863 & 2698 \\ 494 & 533 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2509 & 1596 \\ 798 & 4561 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 4966 & 5007 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4618 & 2907 \\ 3477 & 1312 \end{array}\right),\left(\begin{array}{rr} 5005 & 12 \\ 5004 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[5016])$ is a degree-$1248141312000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5016\Z)$.
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 | $4$ | \( 11913 = 3 \cdot 11 \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 7942 = 2 \cdot 11 \cdot 19^{2} \) |
| $5$ | good | $2$ | \( 202521 = 3 \cdot 11 \cdot 17 \cdot 19^{2} \) |
| $11$ | split multiplicative | $12$ | \( 36822 = 2 \cdot 3 \cdot 17 \cdot 19^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 23826 = 2 \cdot 3 \cdot 11 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 405042.s
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1122.j1, its twist by $-19$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.