Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1732509x+878129500\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1732509xz^2+878129500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-27720147x+56172567854\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 83655 \) | = | $3^{2} \cdot 5 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $26253173966046885$ | = | $3^{13} \cdot 5 \cdot 11^{7} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{4668056654282578921}{213092214885} \) | = | $3^{-7} \cdot 5^{-1} \cdot 7^{3} \cdot 11^{-7} \cdot 13 \cdot 59^{3} \cdot 1721^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2264059425162008375204362763$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2496082386052232024805657509$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9920503162130363$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.82677286560261$ | |||
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: | $\Omega$ | ≈ | $0.35399449071918925004005670772$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 14 $ = $ 2\cdot1\cdot7\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.9559228700686495005607939081 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 4.955922870 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.353994 \cdot 1.000000 \cdot 14}{1^2} \\ & \approx 4.955922870\end{aligned}$$
Modular invariants
Modular form 83655.2.a.bg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1580544 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $13$ | $1$ | $II$ | additive | 1 | 2 | 2 | 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 |
|---|---|---|
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 30030 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 10928 & 7 \\ 5453 & 30024 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4625 & 14 \\ 23086 & 29991 \end{array}\right),\left(\begin{array}{rr} 18026 & 7 \\ 18011 & 30024 \end{array}\right),\left(\begin{array}{rr} 30017 & 14 \\ 30016 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 20012 & 30023 \\ 20027 & 6 \end{array}\right),\left(\begin{array}{rr} 3 & 21458 \\ 30016 & 17123 \end{array}\right)$.
The torsion field $K:=\Q(E[30030])$ is a degree-$1004293914624000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/30030\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 9295 = 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 16731 = 3^{2} \cdot 11 \cdot 13^{2} \) |
| $7$ | good | $2$ | \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \) |
| $11$ | split multiplicative | $12$ | \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $38$ | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 83655.bg
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 27885.g1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.27885.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.128299582125.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.168488679177.1 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.18.132400100347022377560359170818313790183015625.1 | \(\Z/14\Z\) | 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 | ord | add | split | ord | split | add | ss | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 1 | 2 | 1 | - | 0,0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-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
All $p$-adic regulators are identically $1$ since the rank is $0$.