Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-13116x+582547\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-13116xz^2+582547z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1062423x+421489521\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(261/4, 133/8)$ | $0.52214408335119397316248837928$ | $\infty$ |
| $(67, 7)$ | $0.91961216554276408135244454422$ | $\infty$ |
Integral points
\((-101,\pm 931)\), \((51,\pm 209)\), \((67,\pm 7)\)
Invariants
| Conductor: | $N$ | = | \( 40432 \) | = | $2^{4} \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $5006411536$ | = | $2^{4} \cdot 7^{4} \cdot 19^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{119681400064}{2401} \) | = | $2^{8} \cdot 7^{-4} \cdot 19^{2} \cdot 109^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.98179156521946221738259417151$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.23073715468933303909282567961$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9683041695714502$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.776470312294011$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.47108882834945310271032561771$ |
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| Real period: | $\Omega$ | ≈ | $1.2584608425173543430116103380$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot2^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.1141621259019928780062733260 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.114162126 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.258461 \cdot 0.471089 \cdot 12}{1^2} \\ & \approx 7.114162126\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 39168 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $19$ | $3$ | $IV$ | additive | 1 | 2 | 4 | 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 |
|---|---|---|
| $2$ | 2Cn | 4.4.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 76 = 2^{2} \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 68 & 71 \end{array}\right),\left(\begin{array}{rr} 63 & 1 \\ 55 & 0 \end{array}\right),\left(\begin{array}{rr} 35 & 72 \\ 30 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 73 & 4 \\ 72 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[76])$ is a degree-$984960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/76\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 361 = 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $128$ | \( 112 = 2^{4} \cdot 7 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 40432.e consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 20216.i1, its twist by $-4$.
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 |
|---|---|---|---|
| $3$ | 3.3.361.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | 12.4.4452139149819904.3 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | add | ord | ord | split | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 4 | 2 | 3 | 2 | 2 | 2 | - | 2 | 2 | 4 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.