Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1878x-30988\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1878xz^2-30988z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-30051x-2013282\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 774 \) | = | $2 \cdot 3^{2} \cdot 43$ |
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| Discriminant: | $\Delta$ | = | $-3466727424$ | = | $-1 \cdot 2^{12} \cdot 3^{9} \cdot 43 $ |
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| j-invariant: | $j$ | = | \( -\frac{37226247219}{176128} \) | = | $-1 \cdot 2^{-12} \cdot 3^{6} \cdot 7^{3} \cdot 43^{-1} \cdot 53^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.68029274441376825809463392341$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.14366647208731401045180000428$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9571334652445773$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.147043975467183$ | |||
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: | $\Omega$ | ≈ | $0.36250324941495602504620962557$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.4500129976598241001848385023 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.450012998 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.362503 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.450012998\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 576 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $43$ | $1$ | $I_{1}$ | 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.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 516 = 2^{2} \cdot 3 \cdot 43 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 433 & 6 \\ 267 & 19 \end{array}\right),\left(\begin{array}{rr} 42 & 467 \\ 341 & 282 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 511 & 6 \\ 510 & 7 \end{array}\right),\left(\begin{array}{rr} 259 & 6 \\ 261 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[516])$ is a degree-$961196544$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/516\Z)$.
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 | $4$ | \( 129 = 3 \cdot 43 \) |
| $3$ | additive | $2$ | \( 43 \) |
| $43$ | split multiplicative | $44$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 774a
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-66564.2-e1 |
| $3$ | 3.1.516.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.49923.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.137388096.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.7476917787.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.798768.1 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.1027239498704463552.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.29571032522394754282373001818112.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.3165662963095792726583468064289370112.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.8711904474439621583557704112924346548224.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 43 |
|---|---|---|---|
| Reduction type | nonsplit | add | split |
| $\lambda$-invariant(s) | 3 | - | 1 |
| $\mu$-invariant(s) | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.