Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-455745x-1314054591\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-455745xz^2-1314054591z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-36915372x-958056542928\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 21696 \) | = | $2^{6} \cdot 3 \cdot 113$ |
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| Minimal Discriminant: | $\Delta$ | = | $-740065693279926091776$ | = | $-1 \cdot 2^{20} \cdot 3 \cdot 113^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{39934705050538129}{2823126576537804} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-1} \cdot 13^{3} \cdot 113^{-7} \cdot 26293^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6841680480087961411798697543$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6444472771688781770540215721$ |
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| $abc$ quality: | $Q$ | ≈ | $1.055383581120981$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.560019309501589$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.070561465022959277383011682349$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 14 $ = $ 2\cdot1\cdot7 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.98786051032142988336216355289 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.987860510 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.070561 \cdot 1.000000 \cdot 14}{1^2} \\ & \approx 0.987860510\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 677376 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}^{*}$ | additive | 1 | 6 | 20 | 2 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $113$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
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 | $\ell$-adic index |
|---|---|---|---|
| $7$ | 7B.6.3 | 7.24.0.2 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18984 = 2^{3} \cdot 3 \cdot 7 \cdot 113 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 9491 & 0 \\ 0 & 18983 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 18971 & 14 \\ 18970 & 15 \end{array}\right),\left(\begin{array}{rr} 3 & 3398 \\ 4732 & 2675 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4544 & 7 \\ 3143 & 4740 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 15812 & 18977 \\ 15827 & 14244 \end{array}\right),\left(\begin{array}{rr} 14237 & 0 \\ 0 & 18983 \end{array}\right),\left(\begin{array}{rr} 14251 & 18970 \\ 14252 & 18969 \end{array}\right)$.
The torsion field $K:=\Q(E[18984])$ is a degree-$250190657224704$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18984\Z)$.
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 | $2$ | \( 339 = 3 \cdot 113 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 7232 = 2^{6} \cdot 113 \) |
| $7$ | good | $2$ | \( 192 = 2^{6} \cdot 3 \) |
| $113$ | split multiplicative | $114$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 21696h
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 678d2, its twist by $8$.
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 |
|---|---|---|---|
| $3$ | 3.1.339.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.38958219.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.8605184.1 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $14$ | 14.2.20148621553841001725952.1 | \(\Z/7\Z\) | not in database |
| $18$ | 18.0.967116231656834653949477123568697344.1 | \(\Z/14\Z\) | 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 | 113 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord | split |
| $\lambda$-invariant(s) | - | 0 | 6 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 1 | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.