Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+342x+2743\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+342xz^2+2743z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+443205x+121337622\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1, 49)$ | $0.096548555477260134650442281716$ | $\infty$ |
Integral points
\( \left(-7, 11\right) \), \( \left(-7, -5\right) \), \( \left(-1, 49\right) \), \( \left(-1, -49\right) \), \( \left(13, 91\right) \), \( \left(13, -105\right) \), \( \left(97, 931\right) \), \( \left(97, -1029\right) \), \( \left(883, 25821\right) \), \( \left(883, -26705\right) \)
Invariants
| Conductor: | $N$ | = | \( 1274 \) | = | $2 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-5481502208$ | = | $-1 \cdot 2^{9} \cdot 7^{7} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{37595375}{46592} \) | = | $2^{-9} \cdot 5^{3} \cdot 7^{-1} \cdot 13^{-1} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.55479995907208432292161525616$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.41815511545557232963106111556$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8708284255481575$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.090985607549964$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.096548555477260134650442281716$ |
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| Real period: | $\Omega$ | ≈ | $0.90843259190521391125199783964$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ 3^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.1574827618888234043092070422 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.157482762 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.908433 \cdot 0.096549 \cdot 36}{1^2} \\ & \approx 3.157482762\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 576 |
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| $ \Gamma_0(N) $-optimal: | 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $13$ | $1$ | $I_{1}$ | nonsplit 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 | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1856 & 6543 \\ 1737 & 6476 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6535 & 18 \\ 6534 & 19 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1649 & 3294 \\ 3114 & 4285 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 3267 & 6544 \end{array}\right),\left(\begin{array}{rr} 6064 & 9 \\ 3663 & 76 \end{array}\right),\left(\begin{array}{rr} 1639 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right)$.
The torsion field $K:=\Q(E[6552])$ is a degree-$2191186722816$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6552\Z)$.
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 | $4$ | \( 637 = 7^{2} \cdot 13 \) |
| $3$ | good | $2$ | \( 637 = 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 26 = 2 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 98 = 2 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 1274.j
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 182.d3, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/3\Z\) | 2.0.7.1-4732.2-g3 |
| $3$ | 3.1.728.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.385828352.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.12960667629.4 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.480024727.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.3709888.2 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.120028742912169.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.96451446881013167533895415329702805504.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.4900241166540322488131657538469888.1 | \(\Z/18\Z\) | 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 | ord | ss | add | ord | nonsplit | ss | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 1 | 3,1 | - | 3 | 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 | 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.