Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-332258679x-2331026111907\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-332258679xz^2-2331026111907z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5316138867x-149190987300914\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{8900892597}{418609}, \frac{123335039258943}{270840023}\right) \) | $17.930940348982959770446491783$ | $\infty$ |
| \( \left(-\frac{42093}{4}, \frac{42093}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([5758877510259:123335039258943:270840023]\) | $17.930940348982959770446491783$ | $\infty$ |
| \([-84186:42093:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{35603151779}{418609}, \frac{1009715824112580}{270840023}\right) \) | $17.930940348982959770446491783$ | $\infty$ |
| \( \left(-42094, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 471510 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 31$ |
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| Minimal Discriminant: | $\Delta$ | = | $11219047755776040960$ | = | $2^{13} \cdot 3^{10} \cdot 5 \cdot 13^{6} \cdot 31^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{1152829477932246539641}{3188367360} \) | = | $2^{-13} \cdot 3^{-4} \cdot 5^{-1} \cdot 17^{3} \cdot 31^{-2} \cdot 616793^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3135041622623864917948627811$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4817233391975632780704964419$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0264041155005181$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.394941705554956$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $17.930940348982959770446491783$ |
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| Real period: | $\Omega$ | ≈ | $0.035361483071993469094175794788$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{2}\cdot1\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.0725171489238847924507358081 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.072517149 \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 0.035361 \cdot 17.930940 \cdot 32}{2^2} \\ & \approx 5.072517149\end{aligned}$$
Modular invariants
Modular form 471510.2.a.bo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 76677120 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{13}$ | nonsplit multiplicative | 1 | 1 | 13 | 13 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $31$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1240 = 2^{3} \cdot 5 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 2 & 1 \\ 619 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 777 & 466 \\ 464 & 775 \end{array}\right),\left(\begin{array}{rr} 994 & 1 \\ 743 & 0 \end{array}\right),\left(\begin{array}{rr} 1237 & 4 \\ 1236 & 5 \end{array}\right),\left(\begin{array}{rr} 561 & 4 \\ 1122 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[1240])$ is a degree-$54853632000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1240\Z)$.
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 | $4$ | \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 52390 = 2 \cdot 5 \cdot 13^{2} \cdot 31 \) |
| $5$ | split multiplicative | $6$ | \( 94302 = 2 \cdot 3^{2} \cdot 13^{2} \cdot 31 \) |
| $13$ | additive | $86$ | \( 1395 = 3^{2} \cdot 5 \cdot 31 \) |
| $31$ | nonsplit multiplicative | $32$ | \( 15210 = 2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 471510.bo
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 930.j1, its twist by $-39$.
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.