Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-6437424x-6291657232\)
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(homogenize, simplify) |
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\(y^2z=x^3-6437424xz^2-6291657232z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6437424x-6291657232\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4923961/169, 10883367117/2197)$ | $8.1950352069323759341986232224$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 7056 \) | = | $2^{4} \cdot 3^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-27444044051345682432$ | = | $-1 \cdot 2^{12} \cdot 3^{19} \cdot 7^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{1713910976512}{1594323} \) | = | $-1 \cdot 2^{12} \cdot 3^{-13} \cdot 7 \cdot 17^{3} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6527955617333728736692896805$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.11306880413583051515086644495$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1059237401535138$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.618207080988283$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.1950352069323759341986232224$ |
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| Real period: | $\Omega$ | ≈ | $0.047387882733197374935151850991$ |
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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'(E,1)$ | ≈ | $4.6601444085664237676809331407 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.660144409 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.047388 \cdot 8.195035 \cdot 12}{1^2} \\ & \approx 4.660144409\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 174720 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $4$ | $I_{13}^{*}$ | additive | -1 | 2 | 19 | 13 |
| $7$ | $3$ | $IV^{*}$ | 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 |
|---|---|---|
| $13$ | 13B.4.2 | 13.28.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1092 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \), index $336$, genus $9$, and generators
$\left(\begin{array}{rr} 15 & 26 \\ 130 & 287 \end{array}\right),\left(\begin{array}{rr} 14 & 23 \\ 871 & 339 \end{array}\right),\left(\begin{array}{rr} 607 & 1066 \\ 936 & 991 \end{array}\right),\left(\begin{array}{rr} 1067 & 26 \\ 1066 & 27 \end{array}\right),\left(\begin{array}{rr} 545 & 0 \\ 0 & 1091 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 571 & 1066 \\ 572 & 1065 \end{array}\right),\left(\begin{array}{rr} 1078 & 1069 \\ 767 & 753 \end{array}\right),\left(\begin{array}{rr} 727 & 1066 \\ 715 & 753 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1092])$ is a degree-$724598784$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1092\Z)$.
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 | $2$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 784 = 2^{4} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
13.
Its isogeny class 7056bm
consists of 2 curves linked by isogenies of
degree 13.
Twists
The minimal quadratic twist of this elliptic curve is 147c2, its twist by $-84$.
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 |
|---|---|---|---|
| $3$ | 3.1.588.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1037232.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.108884466432.7 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.30849538519502299149815808.1 | \(\Z/13\Z\) | 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 | add | ord | add | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | - | 1 | 5 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 0 | - | 0 | 1 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.