Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+1498x+248501\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+1498xz^2+248501z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+23973x+15928054\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(51, 649)$ | $0.54384049991678506166820349984$ | $\infty$ |
| $(1, 499)$ | $0$ | $3$ |
Integral points
\( \left(-39, 379\right) \), \( \left(-39, -341\right) \), \( \left(1, 499\right) \), \( \left(1, -501\right) \), \( \left(51, 649\right) \), \( \left(51, -701\right) \), \( \left(121, 1419\right) \), \( \left(121, -1541\right) \), \( \left(151, 1899\right) \), \( \left(151, -2051\right) \), \( \left(501, 10999\right) \), \( \left(501, -11501\right) \)
Invariants
| Conductor: | $N$ | = | \( 3330 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-26973000000000$ | = | $-1 \cdot 2^{9} \cdot 3^{6} \cdot 5^{9} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{510273943271}{37000000000} \) | = | $2^{-9} \cdot 5^{-9} \cdot 37^{-1} \cdot 61^{3} \cdot 131^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2567053515417302896903690678$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.70739920720767544399274644934$ |
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| $abc$ quality: | $Q$ | ≈ | $0.996872612230027$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.731087810345289$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.54384049991678506166820349984$ |
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| Real period: | $\Omega$ | ≈ | $0.50953925327506350605374719736$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 162 $ = $ 3^{2}\cdot2\cdot3^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.9879454801100461482519593928 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.987945480 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.509539 \cdot 0.543840 \cdot 162}{3^2} \\ & \approx 4.987945480\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7776 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $37$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $3$ | 3B.1.1 | 9.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13320 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 37 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 6661 & 18 \\ 6669 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9989 & 6642 \\ 0 & 12949 \end{array}\right),\left(\begin{array}{rr} 9991 & 18 \\ 9999 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1087 & 18 \\ 11385 & 12973 \end{array}\right),\left(\begin{array}{rr} 13303 & 18 \\ 13302 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10657 & 18 \\ 2673 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[13320])$ is a degree-$36273255874560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13320\Z)$.
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 | $4$ | \( 1665 = 3^{2} \cdot 5 \cdot 37 \) |
| $3$ | additive | $2$ | \( 37 \) |
| $5$ | split multiplicative | $6$ | \( 666 = 2 \cdot 3^{2} \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 3330.v
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 370.a3, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.1480.1 | \(\Z/6\Z\) | not in database |
| $3$ | 3.3.110889.2 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.3241792000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.50602347.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.26946027.1 | \(\Z/9\Z\) | not in database |
| $9$ | 9.3.3228844269788073792000.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.50198942259523899975028947826347.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.726566512595229689293941092352000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.150193475160708469172913491816448000000.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.24687430834310485750192210349605029937152000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | split | add | split | ord | ord | ord | ord | ord | ord | ord | ord | split | ord | ord | ord |
| $\lambda$-invariant(s) | 12 | - | 4 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | - | 0 | 0 | 0 | 0 | 0 | 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.