Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-3970519034x+96298007745332\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-3970519034xz^2+96298007745332z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5145792667443x+4492895286744223758\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{1425229276}{38025}, \frac{2572142951741}{7414875}\right) \) | $16.411240003603823257756042752$ | $\infty$ |
| \( \left(\frac{145431}{4}, -\frac{145435}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([277919708820:2572142951741:7414875]\) | $16.411240003603823257756042752$ | $\infty$ |
| \([290862:-145435:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{5700929779}{4225}, \frac{21688852108708}{274625}\right) \) | $16.411240003603823257756042752$ | $\infty$ |
| \( \left(1308882, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 490110 \) | = | $2 \cdot 3 \cdot 5 \cdot 17 \cdot 31^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $4471733957024840246775000$ | = | $2^{3} \cdot 3^{4} \cdot 5^{5} \cdot 17^{4} \cdot 31^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{261824518994262912199}{169130025000} \) | = | $2^{-3} \cdot 3^{-4} \cdot 5^{-5} \cdot 17^{-4} \cdot 6397399^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.0483396141044195506734774630$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4728492107405598662266042196$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9975450251705809$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.947015769861304$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $16.411240003603823257756042752$ |
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| Real period: | $\Omega$ | ≈ | $0.064051474195326007819456540258$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.2046564624165287398626193289 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.204656462 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.064051 \cdot 16.411240 \cdot 16}{2^2} \\ & \approx 4.204656462\end{aligned}$$
Modular invariants
Modular form 490110.2.a.ba
For more coefficients, see the Downloads section to the right.
| Modular degree: | 384737280 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $17$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $31$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
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 | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1240 = 2^{3} \cdot 5 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 994 & 1 \\ 743 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 619 & 0 \end{array}\right),\left(\begin{array}{rr} 1237 & 4 \\ 1236 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1004 & 1 \\ 679 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 156 & 1089 \\ 465 & 776 \end{array}\right)$.
The torsion field $K:=\Q(E[1240])$ is a degree-$54853632000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1240\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 155 = 5 \cdot 31 \) |
| $3$ | split multiplicative | $4$ | \( 81685 = 5 \cdot 17 \cdot 31^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 98022 = 2 \cdot 3 \cdot 17 \cdot 31^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 28830 = 2 \cdot 3 \cdot 5 \cdot 31^{2} \) |
| $31$ | additive | $272$ | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 490110ba
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 490110h2, its twist by $-31$.
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.