Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-1647723x+784319578\)
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(homogenize, simplify) |
\(y^2z=x^3-1647723xz^2+784319578z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-1647723x+784319578\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(854, 0)$ | $0$ | $2$ |
Integral points
\( \left(854, 0\right) \)
Invariants
Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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Discriminant: | $\Delta$ | = | $20559497739056640000$ | = | $2^{12} \cdot 3^{7} \cdot 5^{4} \cdot 7^{10} \cdot 13 $ |
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j-invariant: | $j$ | = | \( \frac{1408317602329}{58524375} \) | = | $3^{-1} \cdot 5^{-4} \cdot 7^{-4} \cdot 11^{3} \cdot 13^{-1} \cdot 1019^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4709070567956562610548521021$ |
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.25549865737399945338732099046$ |
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$abc$ quality: | $Q$ | ≈ | $0.8969869824853101$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.185197175220683$ |
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: | $\Omega$ | ≈ | $0.21382888048001440900830908776$ |
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Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2}\cdot1 $ |
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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Special value: | $ L(E,1)$ | ≈ | $3.4212620876802305441329454041 $ |
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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.421262088 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.213829 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 3.421262088\end{aligned}$$
Modular invariants
Modular form 458640.2.a.df
For more coefficients, see the Downloads section to the right.
Modular degree: | 9437184 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
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$2$ | $2$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
$3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
$5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
$7$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
$13$ | $1$ | $I_{1}$ | nonsplit 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 |
---|---|---|
$2$ | 2B | 4.6.0.1 |
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 $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3119 & 10912 \\ 1556 & 10887 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 10913 & 8 \\ 10912 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 8737 & 8 \\ 2188 & 33 \end{array}\right),\left(\begin{array}{rr} 4099 & 4098 \\ 1378 & 6835 \end{array}\right),\left(\begin{array}{rr} 5884 & 1 \\ 9263 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 10914 & 10915 \end{array}\right),\left(\begin{array}{rr} 3632 & 10917 \\ 3635 & 10918 \end{array}\right),\left(\begin{array}{rr} 9552 & 4087 \\ 9527 & 9480 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$38954430627840$ 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 |
---|---|---|---|
$2$ | additive | $2$ | \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \) |
$3$ | additive | $8$ | \( 50960 = 2^{4} \cdot 5 \cdot 7^{2} \cdot 13 \) |
$5$ | nonsplit multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
$7$ | additive | $32$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
$13$ | nonsplit multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 458640.df
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1365.e2, its twist by $-84$.
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 $0$.