Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-1428407x+656494326\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-1428407xz^2+656494326z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1851216147x+30657167512686\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1314, 16992)$ | $4.6121844755867046058584730514$ | $\infty$ |
| $(622, 2714)$ | $0$ | $4$ |
Integral points
\( \left(-1314, 16992\right) \), \( \left(-1314, -15678\right) \), \( \left(622, 2714\right) \), \( \left(622, -3336\right) \)
Invariants
| Conductor: | $N$ | = | \( 121605 \) | = | $3 \cdot 5 \cdot 11^{2} \cdot 67$ |
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| Discriminant: | $\Delta$ | = | $1530047410546875$ | = | $3 \cdot 5^{8} \cdot 11^{7} \cdot 67 $ |
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| j-invariant: | $j$ | = | \( \frac{181938238527312721}{863671875} \) | = | $3^{-1} \cdot 5^{-8} \cdot 11^{-1} \cdot 67^{-1} \cdot 83^{3} \cdot 6827^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1153170566471670355961623769$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.91636942024798176356519058792$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9216176753334842$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.623108088772659$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6121844755867046058584730514$ |
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| Real period: | $\Omega$ | ≈ | $0.42124260967598899070609365986$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{3}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8856972496064523649726203863 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.885697250 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.421243 \cdot 4.612184 \cdot 32}{4^2} \\ & \approx 3.885697250\end{aligned}$$
Modular invariants
Modular form 121605.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2826240 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $11$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $67$ | $1$ | $I_{1}$ | 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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17688 = 2^{3} \cdot 3 \cdot 11 \cdot 67 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11059 & 11058 \\ 15490 & 6643 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5900 & 1 \\ 11815 & 6 \end{array}\right),\left(\begin{array}{rr} 11248 & 17685 \\ 4819 & 17686 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 17682 & 17683 \end{array}\right),\left(\begin{array}{rr} 17681 & 8 \\ 17680 & 9 \end{array}\right),\left(\begin{array}{rr} 15480 & 6641 \\ 15505 & 15552 \end{array}\right),\left(\begin{array}{rr} 17428 & 1 \\ 2927 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[17688])$ is a degree-$402380321587200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17688\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 24321 = 3 \cdot 11^{2} \cdot 67 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 40535 = 5 \cdot 11^{2} \cdot 67 \) |
| $5$ | split multiplicative | $6$ | \( 24321 = 3 \cdot 11^{2} \cdot 67 \) |
| $11$ | additive | $72$ | \( 1005 = 3 \cdot 5 \cdot 67 \) |
| $67$ | nonsplit multiplicative | $68$ | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 121605f
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 11055d3, its twist by $-11$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2211}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.4280496.4 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 67 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | nonsplit | split | ord | add | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 3 | 1 | 4 | 1 | - | 3 | 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 |
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.