Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+44511x-2622548\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+44511xz^2-2622548z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+57686877x-122530648482\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(537383251379/1875409636, 459653750178741135/81216489696616)$ | $26.660656893387269256770982117$ | $\infty$ |
| $(140, 2451)$ | $0$ | $3$ |
Integral points
\( \left(140, 2451\right) \), \( \left(140, -2592\right) \)
Invariants
| Conductor: | $N$ | = | \( 50430 \) | = | $2 \cdot 3 \cdot 5 \cdot 41^{2}$ |
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| Discriminant: | $\Delta$ | = | $-8623719247450680$ | = | $-1 \cdot 2^{3} \cdot 3^{3} \cdot 5 \cdot 41^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{1221431}{1080} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{-1} \cdot 31^{3} \cdot 41$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7452219490980631112522847180$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.73049276203814209132555753069$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8376946671757945$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.037932323517822$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $26.660656893387269256770982117$ |
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| Real period: | $\Omega$ | ≈ | $0.22687606789643638052923774900$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 9 $ = $ 1\cdot3\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.0486650035077247248861691742 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.048665004 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.226876 \cdot 26.660657 \cdot 9}{3^2} \\ & \approx 6.048665004\end{aligned}$$
Modular invariants
Modular form 50430.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 371952 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $41$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 97 & 6 \\ 51 & 19 \end{array}\right),\left(\begin{array}{rr} 61 & 6 \\ 63 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 115 & 6 \\ 114 & 7 \end{array}\right),\left(\begin{array}{rr} 31 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 6 & 1 \\ 43 & 54 \end{array}\right),\left(\begin{array}{rr} 117 & 118 \\ 110 & 113 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$2211840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $4$ | \( 25215 = 3 \cdot 5 \cdot 41^{2} \) |
| $3$ | split multiplicative | $4$ | \( 8405 = 5 \cdot 41^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 10086 = 2 \cdot 3 \cdot 41^{2} \) |
| $41$ | additive | $602$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 50430l
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 50430d1, its twist by $41$.
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 |
|---|---|---|---|
| $3$ | 3.1.201720.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.4882915008000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.47684716875.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.42599252496509116875.3 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.710587487331372225011716800000000000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit | ord | ss | ord | ord | ord | ord | ord | ord | ord | add | ord | ord |
| $\lambda$-invariant(s) | 6 | 2 | 1 | 1 | 3,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | - | 1 | 1 |
| $\mu$-invariant(s) | 0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.