Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-2952497938071x-1952534016650280171\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-2952497938071xz^2-1952534016650280171z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-3826437327740691x-91097369684275555551186\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(-3996917/4, 3996917/8)$ | $0$ | $2$ |
Integral points
None
Invariants
Conductor: | $N$ | = | \( 459186 \) | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 29^{2}$ |
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Discriminant: | $\Delta$ | = | $259280043562354103430949631587388352$ | = | $2^{6} \cdot 3 \cdot 7^{6} \cdot 13^{16} \cdot 29^{7} $ |
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j-invariant: | $j$ | = | \( \frac{4785248917621471258548792495897697}{435894213304313439033688512} \) | = | $2^{-6} \cdot 3^{-1} \cdot 7^{-6} \cdot 13^{-16} \cdot 29^{-1} \cdot 27407^{3} \cdot 6148559^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $5.8344190340143143326407170591$ |
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $4.1507711190210773190490810429$ |
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$abc$ quality: | $Q$ | ≈ | $1.0438156162524606$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.498123933417086$ |
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: | $\Omega$ | ≈ | $0.0036421223224314087133076626871$ |
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Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot1\cdot2\cdot2^{4}\cdot2 $ |
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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Special value: | $ L(E,1)$ | ≈ | $0.46619165727122031530338082394 $ |
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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 0.466191657 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.003642 \cdot 1.000000 \cdot 128}{2^2} \\ & \approx 0.466191657\end{aligned}$$
Modular invariants
Modular form 459186.2.a.e
For more coefficients, see the Downloads section to the right.
Modular degree: | 12386304000 |
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$ \Gamma_0(N) $-optimal: | no | |
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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$2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
$3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
$7$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
$13$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
$29$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | 16.24.0.13 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9744 = 2^{4} \cdot 3 \cdot 7 \cdot 29 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 2797 & 16 \\ 1136 & 9429 \end{array}\right),\left(\begin{array}{rr} 3662 & 2437 \\ 6169 & 4882 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9729 & 16 \\ 9728 & 17 \end{array}\right),\left(\begin{array}{rr} 6712 & 9743 \\ 2945 & 9734 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 2180 & 2121 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 9740 & 9741 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3256 & 1 \\ 6575 & 10 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 9646 & 9731 \end{array}\right)$.
The torsion field $K:=\Q(E[9744])$ is a degree-$8448450232320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9744\Z)$.
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 | $4$ | \( 2523 = 3 \cdot 29^{2} \) |
$3$ | nonsplit multiplicative | $4$ | \( 10933 = 13 \cdot 29^{2} \) |
$7$ | nonsplit multiplicative | $8$ | \( 65598 = 2 \cdot 3 \cdot 13 \cdot 29^{2} \) |
$13$ | split multiplicative | $14$ | \( 35322 = 2 \cdot 3 \cdot 7 \cdot 29^{2} \) |
$29$ | additive | $450$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 459186e
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 15834q5, its twist by $29$.
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 $0$.