Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-439738383x-3197359035738\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-439738383xz^2-3197359035738z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-35618809050x-2330981593480125\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-15098, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-15098:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-135885, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-15098, 0\right) \)
\([-15098:0:1]\)
\( \left(-15098, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 445200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 53$ |
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| Minimal Discriminant: | $\Delta$ | = | $1025270962715148925781250000$ | = | $2^{4} \cdot 3^{7} \cdot 5^{26} \cdot 7 \cdot 53^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{37615499197072174665988096}{4101083850860595703125} \) | = | $2^{11} \cdot 3^{-7} \cdot 5^{-20} \cdot 7^{-1} \cdot 11^{3} \cdot 53^{-2} \cdot 2398573^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9166015346241935442619980399$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.8808335182204949204892076661$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0198257562077644$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.483403451829169$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
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: | $\Omega$ | ≈ | $0.033201770345768707805770782526$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot1\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3905274648953469620154963419 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $36$ = $6^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.390527465 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{36 \cdot 0.033202 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 2.390527465\end{aligned}$$
Modular invariants
Modular form 445200.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 266649600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $3$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $5$ | $4$ | $I_{20}^{*}$ | additive | 1 | 2 | 26 | 20 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $53$ | $2$ | $I_{2}$ | split 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 | 8.12.0.6 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 568 & 3 \\ 565 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 833 & 8 \\ 832 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 248 & 3 \\ 365 & 2 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 834 & 835 \end{array}\right),\left(\begin{array}{rr} 503 & 832 \\ 332 & 807 \end{array}\right),\left(\begin{array}{rr} 739 & 738 \\ 538 & 115 \end{array}\right),\left(\begin{array}{rr} 727 & 732 \\ 730 & 313 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $2$ | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 148400 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 53 \) |
| $5$ | additive | $14$ | \( 17808 = 2^{4} \cdot 3 \cdot 7 \cdot 53 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 21200 = 2^{4} \cdot 5^{2} \cdot 53 \) |
| $53$ | split multiplicative | $54$ | \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 445200.bb
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 44520.j3, its twist by $-20$.
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 $0$.