Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+2509803x-3432138020\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+2509803xz^2-3432138020z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3252705309x-160139589565410\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2468, 132168)$ | $0.93732067659093958912657539595$ | $\infty$ |
| $(3943/4, -3947/8)$ | $0$ | $2$ |
Integral points
\( \left(2468, 132168\right) \), \( \left(2468, -134637\right) \), \( \left(4646, 327099\right) \), \( \left(4646, -331746\right) \), \( \left(38042, 7407051\right) \), \( \left(38042, -7445094\right) \)
Invariants
| Conductor: | $N$ | = | \( 54978 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-6101205432574235702796$ | = | $-1 \cdot 2^{2} \cdot 3^{5} \cdot 7^{6} \cdot 11^{12} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{14861225463775641287}{51859390496937804} \) | = | $2^{-2} \cdot 3^{-5} \cdot 11^{-12} \cdot 17^{-1} \cdot 2458583^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8632210636683464724587195279$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8902659891406898199060431562$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0924697760886544$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.262281785455576$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.93732067659093958912657539595$ |
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| Real period: | $\Omega$ | ≈ | $0.068415328756343320049661556613$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 240 $ = $ 2\cdot5\cdot2\cdot( 2^{2} \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8476261343452371781359734301 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.847626134 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.068415 \cdot 0.937321 \cdot 240}{2^2} \\ & \approx 3.847626134\end{aligned}$$
Modular invariants
Modular form 54978.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4423680 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $17$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 31416 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 22849 & 26936 \\ 10612 & 13497 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31409 & 8 \\ 31408 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 21883 & 21882 \\ 16282 & 18523 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 31410 & 31411 \end{array}\right),\left(\begin{array}{rr} 10480 & 17955 \\ 19453 & 26930 \end{array}\right),\left(\begin{array}{rr} 19643 & 15148 \\ 28658 & 12909 \end{array}\right),\left(\begin{array}{rr} 22439 & 0 \\ 0 & 31415 \end{array}\right),\left(\begin{array}{rr} 8716 & 26929 \\ 24311 & 8982 \end{array}\right)$.
The torsion field $K:=\Q(E[31416])$ is a degree-$3201968583475200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/31416\Z)$.
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 | $4$ | \( 2499 = 3 \cdot 7^{2} \cdot 17 \) |
| $3$ | split multiplicative | $4$ | \( 1666 = 2 \cdot 7^{2} \cdot 17 \) |
| $5$ | good | $2$ | \( 18326 = 2 \cdot 7^{2} \cdot 11 \cdot 17 \) |
| $7$ | additive | $26$ | \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \) |
| $11$ | split multiplicative | $12$ | \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 54978.q
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1122.c4, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-51}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{119}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-21}, \sqrt{-51})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | ord | add | split | ord | split | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 10 | 2 | 1 | - | 2 | 1 | 2 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 2 | 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.