Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+1036500x-941481250\)
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(homogenize, simplify) |
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\(y^2z=x^3+1036500xz^2-941481250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1036500x-941481250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10999, 1158057\right) \) | $2.8476810949573728371751639120$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([10999:1158057:1]\) | $2.8476810949573728371751639120$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10999, 1158057\right) \) | $2.8476810949573728371751639120$ | $\infty$ |
Integral points
\((10999,\pm 1158057)\)
\([10999:\pm 1158057:1]\)
\((10999,\pm 1158057)\)
Invariants
| Conductor: | $N$ | = | \( 417600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $-454186063987875000000$ | = | $-1 \cdot 2^{6} \cdot 3^{11} \cdot 5^{9} \cdot 29^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{1351431663616}{4984209207} \) | = | $2^{12} \cdot 3^{-5} \cdot 29^{-5} \cdot 691^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6466176591000153904851881770$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.54365949016041260912837999789$ |
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| $abc$ quality: | $Q$ | ≈ | $1.015700664847297$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.237976494091972$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.8476810949573728371751639120$ |
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| Real period: | $\Omega$ | ≈ | $0.084779964618100380333448014597$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 1\cdot2^{2}\cdot2\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.6570520989647767412724039972 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.657052099 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.084780 \cdot 2.847681 \cdot 40}{1^2} \\ & \approx 9.657052099\end{aligned}$$
Modular invariants
Modular form 417600.2.a.fb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14592000 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 6 | 6 | 0 |
| $3$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $29$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 |
|---|---|---|---|
| $5$ | 5B.4.1 | 5.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 3474 & 3467 \\ 2665 & 2729 \end{array}\right),\left(\begin{array}{rr} 3449 & 3470 \\ 3325 & 3429 \end{array}\right),\left(\begin{array}{rr} 2619 & 3470 \\ 2620 & 3469 \end{array}\right),\left(\begin{array}{rr} 2899 & 3470 \\ 3185 & 3429 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 3425 & 3361 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2609 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 1739 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 3471 & 10 \\ 3470 & 11 \end{array}\right),\left(\begin{array}{rr} 3479 & 3470 \\ 3475 & 2733 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$502883942400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
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 | $2$ | \( 1305 = 3^{2} \cdot 5 \cdot 29 \) |
| $3$ | additive | $8$ | \( 46400 = 2^{6} \cdot 5^{2} \cdot 29 \) |
| $5$ | additive | $14$ | \( 576 = 2^{6} \cdot 3^{2} \) |
| $29$ | split multiplicative | $30$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 417600fb
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 2175j2, its twist by $120$.
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.