Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-971995140x+8667440542800\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-971995140xz^2+8667440542800z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15551922243x+554700642816958\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{100035}{4}, -\frac{100035}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([200070:-100035:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(100034, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 446490 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 41$ |
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| Minimal Discriminant: | $\Delta$ | = | $26320266182238806906742240000$ | = | $2^{8} \cdot 3^{30} \cdot 5^{4} \cdot 11^{7} \cdot 41 $ |
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| j-invariant: | $j$ | = | \( \frac{78637980788709358357081}{20380115352468960000} \) | = | $2^{-8} \cdot 3^{-24} \cdot 5^{-4} \cdot 11^{-1} \cdot 13^{3} \cdot 41^{-1} \cdot 3295597^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1624713019698851365945341411$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4142175212366450188659397337$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9919247334093735$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.665094251917492$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
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: | $\Omega$ | ≈ | $0.035184079097684287479494617406$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.28147263278147429983595693925 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.281472633 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.035184 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 0.281472633\end{aligned}$$
Modular invariants
Modular form 446490.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 448266240 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $4$ | $I_{24}^{*}$ | additive | -1 | 2 | 30 | 24 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $41$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 54120 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 41 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 43297 & 18048 \\ 46908 & 18073 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20299 & 20298 \\ 42858 & 33835 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 54114 & 54115 \end{array}\right),\left(\begin{array}{rr} 29516 & 18039 \\ 16377 & 36074 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 24808 & 51873 \\ 42873 & 24880 \end{array}\right),\left(\begin{array}{rr} 36079 & 0 \\ 0 & 54119 \end{array}\right),\left(\begin{array}{rr} 54113 & 8 \\ 54112 & 9 \end{array}\right),\left(\begin{array}{rr} 30364 & 36081 \\ 16743 & 18046 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[54120])$ is a degree-$26813870899200000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/54120\Z)$.
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 | $4$ | \( 44649 = 3^{2} \cdot 11^{2} \cdot 41 \) |
| $3$ | additive | $6$ | \( 49610 = 2 \cdot 5 \cdot 11^{2} \cdot 41 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 89298 = 2 \cdot 3^{2} \cdot 11^{2} \cdot 41 \) |
| $11$ | additive | $72$ | \( 3690 = 2 \cdot 3^{2} \cdot 5 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 10890 = 2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 446490.g
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 13530.k2, its twist by $33$.
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 $0$.