Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1647365x-1086333699\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1647365xz^2-1086333699z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-26357835x-69551714554\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2207, 76512)$ | $0.61853593455387427506355617598$ | $\infty$ |
| $(1535, -768)$ | $0$ | $2$ |
Integral points
\( \left(1535, -768\right) \), \( \left(2207, 76512\right) \), \( \left(2207, -78720\right) \), \( \left(2559, 105728\right) \), \( \left(2559, -108288\right) \), \( \left(6827, 549600\right) \), \( \left(6827, -556428\right) \), \( \left(9599, 926592\right) \), \( \left(9599, -936192\right) \), \( \left(270335, 140420352\right) \), \( \left(270335, -140690688\right) \)
Invariants
| Conductor: | $N$ | = | \( 126126 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-224078186780527951872$ | = | $-1 \cdot 2^{24} \cdot 3^{8} \cdot 7^{6} \cdot 11^{3} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{5764706497797625}{2612665516032} \) | = | $-1 \cdot 2^{-24} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{3} \cdot 11^{-3} \cdot 13^{-1} \cdot 47^{3} \cdot 109^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6115123473240230317322124076$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0892511284623115334819134174$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9888164482102493$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.69441293866671$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.61853593455387427506355617598$ |
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| Real period: | $\Omega$ | ≈ | $0.065203650509458237955757791815$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1152 $ = $ ( 2^{3} \cdot 3 )\cdot2^{2}\cdot2^{2}\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.615270660407282091150175355 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.615270660 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.065204 \cdot 0.618536 \cdot 1152}{2^2} \\ & \approx 11.615270660\end{aligned}$$
Modular invariants
Modular form 126126.2.a.es
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3981312 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $24$ | $I_{24}$ | split multiplicative | -1 | 1 | 24 | 24 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12012 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 5818 & 6867 \\ 9345 & 3424 \end{array}\right),\left(\begin{array}{rr} 8579 & 0 \\ 0 & 12011 \end{array}\right),\left(\begin{array}{rr} 4222 & 6867 \\ 441 & 3424 \end{array}\right),\left(\begin{array}{rr} 4003 & 8568 \\ 2282 & 3359 \end{array}\right),\left(\begin{array}{rr} 12001 & 12 \\ 12000 & 13 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 11962 & 12003 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1863 & 5866 \\ 1022 & 3305 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[12012])$ is a degree-$33476463820800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12012\Z)$.
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 | $4$ | \( 63063 = 3^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $3$ | additive | $8$ | \( 637 = 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | split multiplicative | $12$ | \( 11466 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 126126.es
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 858.c2, its twist by $21$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-143}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.1009008.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-143})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.1735406945181.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.20819068498964736.29 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1018097144064.6 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.87124237460981265231306482684436623707324674048.1 | \(\Z/18\Z\) | 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 | add | ss | add | split | nonsplit | ss | ord | ss | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 6 | - | 1,1 | - | 2 | 1 | 1,1 | 1 | 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 | 0,0 |
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.