Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+2150217x-5371246694\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+2150217xz^2-5371246694z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2786681853x-250609245789186\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1351, -676)$ | $0$ | $2$ |
Integral points
\( \left(1351, -676\right) \)
Invariants
| Conductor: | $N$ | = | \( 28830 \) | = | $2 \cdot 3 \cdot 5 \cdot 31^{2}$ |
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| Discriminant: | $\Delta$ | = | $-13100406335093760000000$ | = | $-1 \cdot 2^{16} \cdot 3 \cdot 5^{7} \cdot 31^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{1238798620042199}{14760960000000} \) | = | $2^{-16} \cdot 3^{-1} \cdot 5^{-7} \cdot 31^{-2} \cdot 211^{3} \cdot 509^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9248126578973828992358958268$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2078190556548097762713136645$ |
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| $abc$ quality: | $Q$ | ≈ | $1.013256971846666$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.680289043827377$ | |||
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: | $\Omega$ | ≈ | $0.061934765093202969755860807119$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 56 $ = $ 2\cdot1\cdot7\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.4683468452193663063282051987 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.468346845 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.061935 \cdot 1.000000 \cdot 56}{2^2} \\ & \approx 3.468346845\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2580480 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{16}$ | nonsplit multiplicative | 1 | 1 | 16 | 16 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $31$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 374 & 1 \\ 743 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1801 & 4 \\ 1742 & 9 \end{array}\right),\left(\begin{array}{rr} 622 & 1 \\ 619 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1857 & 4 \\ 1856 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 469 & 1396 \\ 464 & 1395 \end{array}\right)$.
The torsion field $K:=\Q(E[1860])$ is a degree-$164560896000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1860\Z)$.
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 | $4$ | \( 14415 = 3 \cdot 5 \cdot 31^{2} \) |
| $3$ | split multiplicative | $4$ | \( 9610 = 2 \cdot 5 \cdot 31^{2} \) |
| $5$ | split multiplicative | $6$ | \( 5766 = 2 \cdot 3 \cdot 31^{2} \) |
| $7$ | good | $2$ | \( 5766 = 2 \cdot 3 \cdot 31^{2} \) |
| $31$ | additive | $512$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 28830.t
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 930.d2, its twist by $-31$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.230640.4 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.11968832160000.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.168311702250000.16 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 31 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | split | split | ord | add |
| $\lambda$-invariant(s) | 16 | 1 | 1 | 4 | - |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.