Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2+2964x+24144\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z+2964xz^2+24144z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+3840669x+1068849054\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(41, 445)$ | $2.4794821359784509089790720672$ | $\infty$ |
| $(17, 274)$ | $4.2098892724692510454872536925$ | $\infty$ |
| $(-8, 4)$ | $0$ | $2$ |
Integral points
\( \left(-8, 4\right) \), \( \left(17, 274\right) \), \( \left(17, -291\right) \), \( \left(41, 445\right) \), \( \left(41, -486\right) \), \( \left(56, 580\right) \), \( \left(56, -636\right) \)
Invariants
| Conductor: | $N$ | = | \( 54978 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $-1892383223808$ | = | $-1 \cdot 2^{12} \cdot 3 \cdot 7^{7} \cdot 11 \cdot 17 $ |
|
| j-invariant: | $j$ | = | \( \frac{24464768327}{16084992} \) | = | $2^{-12} \cdot 3^{-1} \cdot 7^{-1} \cdot 11^{-1} \cdot 17^{-1} \cdot 2903^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0445597977394349363606988113$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.071604723211778283808022439579$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8617725193907065$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.261289698197532$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.396872184542777184403106875$ |
|
| Real period: | $\Omega$ | ≈ | $0.52149603499919968225969139003$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $5.4219276206325257883016733601 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.421927621 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.521496 \cdot 10.396872 \cdot 4}{2^2} \\ & \approx 5.421927621\end{aligned}$$
Modular invariants
Modular form 54978.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 129024 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $17$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 |
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
$\left(\begin{array}{rr} 7 & 6 \\ 31410 & 31411 \end{array}\right),\left(\begin{array}{rr} 20948 & 1 \\ 10495 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22180 & 1 \\ 1871 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 25712 & 3 \\ 19997 & 2 \end{array}\right),\left(\begin{array}{rr} 22432 & 31413 \\ 17947 & 31414 \end{array}\right),\left(\begin{array}{rr} 19643 & 19638 \\ 11786 & 27491 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 27497 & 27492 \\ 27494 & 11783 \end{array}\right),\left(\begin{array}{rr} 31409 & 8 \\ 31408 & 9 \end{array}\right)$.
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 | $4$ | \( 27489 = 3 \cdot 7^{2} \cdot 11 \cdot 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 9163 = 7^{2} \cdot 11 \cdot 17 \) |
| $7$ | additive | $32$ | \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \) |
| $17$ | nonsplit 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 54978h
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 7854i1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3927}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{357}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-11}, \sqrt{357})\) | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | nonsplit | ord | add | nonsplit | ord | nonsplit | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | 2 | 2 | - | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-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.