Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-161990x-13852125\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-161990xz^2-13852125z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-209939067x-645654926826\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-350, 175\right) \) | $0$ | $2$ |
| \( \left(-90, 45\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-350:175:1]\) | $0$ | $2$ |
| \([-90:45:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-12597, 0\right) \) | $0$ | $2$ |
| \( \left(-3237, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-350, 175\right) \), \( \left(-90, 45\right) \)
\([-350:175:1]\), \([-90:45:1]\)
\( \left(-12597, 0\right) \), \( \left(-3237, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 413205 \) | = | $3 \cdot 5 \cdot 13^{2} \cdot 163$ |
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| Minimal Discriminant: | $\Delta$ | = | $189316243546668225$ | = | $3^{10} \cdot 5^{2} \cdot 13^{6} \cdot 163^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{97393143178729}{39221822025} \) | = | $3^{-10} \cdot 5^{-2} \cdot 139^{3} \cdot 163^{-2} \cdot 331^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0131470487377918236887053942$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.73067237000702345566196167342$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9271338051584149$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6808367284339667$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.24643865999704298405766446592$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ ( 2 \cdot 5 )\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.4643865999704298405766446592 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.464386600 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.246439 \cdot 1.000000 \cdot 160}{4^2} \\ & \approx 2.464386600\end{aligned}$$
Modular invariants
Modular form 413205.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4423680 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $163$ | $2$ | $I_{2}$ | 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 |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 127140 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \cdot 163 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 29339 & 0 \\ 0 & 127139 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 122253 & 117364 \\ 107588 & 39131 \end{array}\right),\left(\begin{array}{rr} 50857 & 117364 \\ 33254 & 107589 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 25923 & 58682 \\ 90298 & 68459 \end{array}\right),\left(\begin{array}{rr} 84761 & 58682 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 127137 & 4 \\ 127136 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[127140])$ is a degree-$847242693110661120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/127140\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 169 = 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 137735 = 5 \cdot 13^{2} \cdot 163 \) |
| $5$ | split multiplicative | $6$ | \( 27547 = 13^{2} \cdot 163 \) |
| $13$ | additive | $86$ | \( 2445 = 3 \cdot 5 \cdot 163 \) |
| $163$ | split multiplicative | $164$ | \( 2535 = 3 \cdot 5 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 413205d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2445c2, its twist by $13$.
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 $0$.