Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-1644197x-710860251\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-1644197xz^2-710860251z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-2130879987x-33133932674226\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
Conductor: | $N$ | = | \( 451770 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 37^{2}$ |
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Discriminant: | $\Delta$ | = | $66594913064379021960$ | = | $2^{3} \cdot 3^{13} \cdot 5 \cdot 11 \cdot 37^{7} $ |
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j-invariant: | $j$ | = | \( \frac{191591101730449}{25955578440} \) | = | $2^{-3} \cdot 3^{-13} \cdot 5^{-1} \cdot 11^{-1} \cdot 37^{-1} \cdot 57649^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5309768012154168731819179579$ |
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.72551784489330465099787012239$ |
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$abc$ quality: | $Q$ | ≈ | $0.8965269744433728$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.189554692600231$ |
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: | $\Omega$ | ≈ | $0.13452101328330659679899926903$ |
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Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot1\cdot1\cdot2 $ |
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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Special value: | $ L(E,1)$ | ≈ | $2.4213782390995187423819868425 $ |
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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.421378239 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.134521 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 2.421378239\end{aligned}$$
Modular invariants
Modular form 451770.2.a.o
For more coefficients, see the Downloads section to the right.
Modular degree: | 17072640 |
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 |
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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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$2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
$3$ | $1$ | $I_{13}$ | nonsplit multiplicative | 1 | 1 | 13 | 13 |
$5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
$11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
$37$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48840 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 35521 & 2 \\ 35521 & 3 \end{array}\right),\left(\begin{array}{rr} 32561 & 2 \\ 32561 & 3 \end{array}\right),\left(\begin{array}{rr} 48839 & 2 \\ 48838 & 3 \end{array}\right),\left(\begin{array}{rr} 19537 & 2 \\ 19537 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 48839 & 0 \end{array}\right),\left(\begin{array}{rr} 3961 & 2 \\ 3961 & 3 \end{array}\right),\left(\begin{array}{rr} 24421 & 2 \\ 24421 & 3 \end{array}\right),\left(\begin{array}{rr} 36631 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[48840])$ is a degree-$425606202261504000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48840\Z)$.
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 |
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$2$ | nonsplit multiplicative | $4$ | \( 225885 = 3 \cdot 5 \cdot 11 \cdot 37^{2} \) |
$3$ | nonsplit multiplicative | $4$ | \( 75295 = 5 \cdot 11 \cdot 37^{2} \) |
$5$ | split multiplicative | $6$ | \( 90354 = 2 \cdot 3 \cdot 11 \cdot 37^{2} \) |
$11$ | nonsplit multiplicative | $12$ | \( 41070 = 2 \cdot 3 \cdot 5 \cdot 37^{2} \) |
$13$ | good | $2$ | \( 150590 = 2 \cdot 5 \cdot 11 \cdot 37^{2} \) |
$37$ | additive | $722$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 451770o consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 12210r1, its twist by $37$.
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 $0$.