Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-4155849005x-103117360747003\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-4155849005xz^2-103117360747003z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-66493584075x-6599577581392250\)
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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 |
|---|---|---|
| \( \left(-37161, 18580\right) \) | $0$ | $2$ |
| \( \left(74439, -37220\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-37161:18580:1]\) | $0$ | $2$ |
| \([74439:-37220:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-148645, 0\right) \) | $0$ | $2$ |
| \( \left(297755, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-37161, 18580\right) \), \( \left(74439, -37220\right) \)
\([-37161:18580:1]\), \([74439:-37220:1]\)
\( \left(-148645, 0\right) \), \( \left(297755, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 432450 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $33609882443006471006250000$ | = | $2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 31^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{785209010066844481}{3324675600} \) | = | $2^{-4} \cdot 3^{-2} \cdot 5^{-2} \cdot 31^{-4} \cdot 922561^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1062707547285485357495416990$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0352520519348703797869572517$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0007838605326649$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.014919896982211$ | |||
| 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: | $\Omega$ | ≈ | $0.018803328139465375052166977175$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.8136520037031360133547461567 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
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BSD formula
$$\begin{aligned} 4.813652004 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.018803 \cdot 1.000000 \cdot 256}{4^2} \\ & \approx 4.813652004\end{aligned}$$
Modular invariants
Modular form 432450.2.a.fz
For more coefficients, see the Downloads section to the right.
| Modular degree: | 377487360 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $31$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 | 8.48.0.113 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1865 & 932 \\ 952 & 1869 \end{array}\right),\left(\begin{array}{rr} 2473 & 3718 \\ 18 & 5 \end{array}\right),\left(\begin{array}{rr} 1919 & 3712 \\ 236 & 3687 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 3716 & 3717 \end{array}\right),\left(\begin{array}{rr} 1481 & 3718 \\ 1506 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 936 \\ 3714 & 2785 \end{array}\right),\left(\begin{array}{rr} 3713 & 8 \\ 3712 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3720])$ is a degree-$164560896000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3720\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 216225 = 3^{2} \cdot 5^{2} \cdot 31^{2} \) |
| $3$ | additive | $8$ | \( 48050 = 2 \cdot 5^{2} \cdot 31^{2} \) |
| $5$ | additive | $18$ | \( 17298 = 2 \cdot 3^{2} \cdot 31^{2} \) |
| $31$ | additive | $512$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 432450.fz
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 930.o2, its twist by $465$.
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 $0$.