Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-180208x-32294213\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-180208xz^2-32294213z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-14596875x-23586271875\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{31159}{49}, \frac{3594849}{343}\right) \) | $10.128929499626511555414447457$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([218113:3594849:343]\) | $10.128929499626511555414447457$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{280284}{49}, \frac{97060923}{343}\right) \) | $10.128929499626511555414447457$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 445200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 53$ |
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| Minimal Discriminant: | $\Delta$ | = | $-77671922343750000$ | = | $-1 \cdot 2^{4} \cdot 3^{2} \cdot 5^{10} \cdot 7 \cdot 53^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{4142173600000}{497100303} \) | = | $-1 \cdot 2^{8} \cdot 3^{-2} \cdot 5^{5} \cdot 7^{-1} \cdot 53^{-4} \cdot 173^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9762912399777909594069100662$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.40404391942939221076719991469$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9285570968058968$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.698802694335243$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.128929499626511555414447457$ |
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| Real period: | $\Omega$ | ≈ | $0.11508329978466625171632696769$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.6626825204130694481183280099 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.662682520 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.115083 \cdot 10.128929 \cdot 4}{1^2} \\ & \approx 4.662682520\end{aligned}$$
Modular invariants
Modular form 445200.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4945920 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $53$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 label 14.2.0.a.1, level \( 14 = 2 \cdot 7 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 13 & 2 \\ 12 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 13 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 3 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[14])$ is a degree-$6048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14\Z)$.
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 | $2$ | \( 175 = 5^{2} \cdot 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 148400 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 53 \) |
| $5$ | additive | $2$ | \( 17808 = 2^{4} \cdot 3 \cdot 7 \cdot 53 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 63600 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 53 \) |
| $53$ | nonsplit multiplicative | $54$ | \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 445200n consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 222600cc1, its twist by $-20$.
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.