Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-36445555x-84681498425\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-36445555xz^2-84681498425z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-583128883x-5420199028082\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3470820551/373321, 139555175287800/228099131)$ | $18.036198790893540879009326775$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 2890 \) | = | $2 \cdot 5 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-289521182854003906250$ | = | $-1 \cdot 2 \cdot 5^{13} \cdot 17^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{45145776875761017}{2441406250} \) | = | $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 5^{-13} \cdot 118691^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9930098570959011913427141054$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.86809984905373913115556314199$ |
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| $abc$ quality: | $Q$ | ≈ | $1.082527750146711$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.011995518160992$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $18.036198790893540879009326775$ |
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| Real period: | $\Omega$ | ≈ | $0.030722532146147981069487309065$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.1082353942950843154113825506 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.108235394 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.030723 \cdot 18.036199 \cdot 2}{1^2} \\ & \approx 1.108235394\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 551616 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{13}$ | nonsplit multiplicative | 1 | 1 | 13 | 13 |
| $17$ | $2$ | $III^{*}$ | 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 |
|---|---|---|
| $13$ | 13B.5.2 | 13.42.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8840 = 2^{3} \cdot 5 \cdot 13 \cdot 17 \), index $336$, genus $9$, and generators
$\left(\begin{array}{rr} 2225 & 4446 \\ 8424 & 4889 \end{array}\right),\left(\begin{array}{rr} 8815 & 26 \\ 8814 & 27 \end{array}\right),\left(\begin{array}{rr} 3537 & 26 \\ 1781 & 339 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14 & 23 \\ 871 & 1431 \end{array}\right),\left(\begin{array}{rr} 3099 & 8814 \\ 1131 & 7665 \end{array}\right),\left(\begin{array}{rr} 4421 & 26 \\ 4433 & 339 \end{array}\right),\left(\begin{array}{rr} 6631 & 26 \\ 6643 & 339 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[8840])$ is a degree-$4504934154240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8840\Z)$.
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 | $4$ | \( 85 = 5 \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 578 = 2 \cdot 17^{2} \) |
| $13$ | good | $2$ | \( 578 = 2 \cdot 17^{2} \) |
| $17$ | additive | $98$ | \( 10 = 2 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
13.
Its isogeny class 2890d
consists of 2 curves linked by isogenies of
degree 13.
Twists
The minimal quadratic twist of this elliptic curve is 2890j2, its twist by $17$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.680.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.314432000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.212528495470874611248389.2 | \(\Z/13\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | ss | nonsplit | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 1,1 | 3 | 1 | 1 | 3 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0,0 | 0 | 0 | 0 | 1 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.