Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-410139x-38853851\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-410139xz^2-38853851z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6562227x-2493208690\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-145, 4262)$ | $3.4100414411237134286983002465$ | $\infty$ |
| $(-586, 293)$ | $0$ | $2$ |
Integral points
\( \left(-586, 293\right) \), \( \left(-145, 4262\right) \), \( \left(-145, -4117\right) \), \( \left(822, 12965\right) \), \( \left(822, -13787\right) \)
Invariants
| Conductor: | $N$ | = | \( 126126 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $3759834632592310272$ | = | $2^{14} \cdot 3^{11} \cdot 7^{7} \cdot 11^{2} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{88961427666721}{43838226432} \) | = | $2^{-14} \cdot 3^{-5} \cdot 7^{-1} \cdot 11^{-2} \cdot 13^{-1} \cdot 44641^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2574003114807033919730362851$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.73513909261899189372273729492$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9306445331244925$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.2900128766707475$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4100414411237134286983002465$ |
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| Real period: | $\Omega$ | ≈ | $0.19839421416252463239404686885$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7061299678935285364597666581 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.706129968 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.198394 \cdot 3.410041 \cdot 16}{2^2} \\ & \approx 2.706129968\end{aligned}$$
Modular invariants
Modular form 126126.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3440640 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{14}$ | nonsplit multiplicative | 1 | 1 | 14 | 14 |
| $3$ | $2$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $1$ | $I_{1}$ | split 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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24024 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 21025 & 3004 \\ 9008 & 15015 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 4369 & 4 \\ 8738 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 24021 & 4 \\ 24020 & 5 \end{array}\right),\left(\begin{array}{rr} 12013 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1850 & 1 \\ 11087 & 0 \end{array}\right),\left(\begin{array}{rr} 6866 & 1 \\ 20591 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 16018 & 1 \\ 16015 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[24024])$ is a degree-$4284987369062400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24024\Z)$.
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 | $4$ | \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 14014 = 2 \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $7$ | additive | $32$ | \( 1287 = 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | split multiplicative | $12$ | \( 11466 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $13$ | split 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.
Its isogeny class 126126.c
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 6006.ba2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{273}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.2114112.3 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | ord | add | split | split | ord | ss | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 1 | - | 2 | 2 | 1 | 3,1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 3 |
| $\mu$-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.