Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-8931231x-9459802080\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-8931231xz^2-9459802080z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-11574875403x-441321801218298\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1872, 27396)$ | $2.5689122238197048997267663748$ | $\infty$ |
| $(-1312, 656)$ | $0$ | $2$ |
| $(3420, -1710)$ | $0$ | $2$ |
Integral points
\( \left(-1872, 27396\right) \), \( \left(-1872, -25524\right) \), \( \left(-1312, 656\right) \), \( \left(3420, -1710\right) \), \( \left(16524, 2078550\right) \), \( \left(16524, -2095074\right) \)
Invariants
| Conductor: | $N$ | = | \( 124215 \) | = | $3 \cdot 5 \cdot 7^{2} \cdot 13^{2}$ |
|
| Discriminant: | $\Delta$ | = | $6941985375786779088225$ | = | $3^{10} \cdot 5^{2} \cdot 7^{8} \cdot 13^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{138742439989609}{12224619225} \) | = | $3^{-10} \cdot 5^{-2} \cdot 7^{-2} \cdot 13^{-2} \cdot 51769^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9312703461635598148568788684$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.67584059290513479427745877590$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9312987645826678$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.083543958200801$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5689122238197048997267663748$ |
|
| Real period: | $\Omega$ | ≈ | $0.087822876709249717333003325149$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 320 $ = $ ( 2 \cdot 5 )\cdot2\cdot2^{2}\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.5121852301880491672543781784 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.512185230 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.087823 \cdot 2.568912 \cdot 320}{4^2} \\ & \approx 4.512185230\end{aligned}$$
Modular invariants
Modular form 124215.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7741440 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5460 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5457 & 4 \\ 5456 & 5 \end{array}\right),\left(\begin{array}{rr} 3641 & 4 \\ 1822 & 9 \end{array}\right),\left(\begin{array}{rr} 3777 & 5458 \\ 2942 & 1 \end{array}\right),\left(\begin{array}{rr} 3119 & 5458 \\ 0 & 5459 \end{array}\right),\left(\begin{array}{rr} 3277 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2731 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[5460])$ is a degree-$2434651914240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5460\Z)$.
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 | $2$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 41405 = 5 \cdot 7^{2} \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 2535 = 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 124215.y
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1365.c2, its twist by $-91$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-7}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-65}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | ord | split | nonsplit | add | ss | add | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 2 | 1 | - | 1,1 | - | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 3 | 1 |
| $\mu$-invariant(s) | 1 | 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.