Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2+462112x-54622719\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z+462112xz^2-54622719z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+598897125x-2557461026250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(669, 23213\right) \) | $1.9935513628598670969815262588$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([669:23213:1]\) | $1.9935513628598670969815262588$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(24099, 5086368\right) \) | $1.9935513628598670969815262588$ | $\infty$ |
Integral points
\( \left(669, 23213\right) \), \( \left(669, -23883\right) \)
\([669:23213:1]\), \([669:-23883:1]\)
\((24099,\pm 5086368)\)
Invariants
| Conductor: | $N$ | = | \( 42050 \) | = | $2 \cdot 5^{2} \cdot 29^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-7613738508800000000$ | = | $-1 \cdot 2^{15} \cdot 5^{8} \cdot 29^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{46969655}{32768} \) | = | $2^{-15} \cdot 5 \cdot 211^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3115734194004605138358129213$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.44503310388217674948966265037$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0629647337743247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.7662371736962115$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9935513628598670969815262588$ |
|
| Real period: | $\Omega$ | ≈ | $0.13243141240235847756347514071$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 30 $ = $ ( 3 \cdot 5 )\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $7.9202646804053654727147475393 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 7.920264680 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.132431 \cdot 1.993551 \cdot 30}{1^2} \\ & \approx 7.920264680\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 756000 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $5$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $29$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
| $5$ | 5B.4.1 | 5.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2640 & 1 \end{array}\right),\left(\begin{array}{rr} 2321 & 1160 \\ 2320 & 1161 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 870 & 1 \end{array}\right),\left(\begin{array}{rr} 839 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1740 & 1 \end{array}\right),\left(\begin{array}{rr} 871 & 2610 \\ 2175 & 2611 \end{array}\right),\left(\begin{array}{rr} 1 & 2436 \\ 1740 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 522 \\ 2610 & 1741 \end{array}\right),\left(\begin{array}{rr} 1 & 1392 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3046 & 1305 \\ 3045 & 1 \end{array}\right),\left(\begin{array}{rr} 1741 & 2610 \\ 3045 & 2611 \end{array}\right),\left(\begin{array}{rr} 2319 & 3190 \\ 1160 & 2783 \end{array}\right),\left(\begin{array}{rr} 841 & 2640 \\ 840 & 841 \end{array}\right),\left(\begin{array}{rr} 2321 & 2610 \\ 2320 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$62860492800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 21025 = 5^{2} \cdot 29^{2} \) |
| $3$ | good | $2$ | \( 21025 = 5^{2} \cdot 29^{2} \) |
| $5$ | additive | $14$ | \( 841 = 29^{2} \) |
| $29$ | additive | $422$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 42050.x
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50.b4, its twist by $145$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-87}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{145}) \) | \(\Z/5\Z\) | not in database |
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-87})\) | \(\Z/15\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.11112238125.5 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.26340120000.7 | \(\Z/6\Z\) | not in database |
| $6$ | 6.2.4877800000.2 | \(\Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/15\Z\) | not in database |
| $12$ | deg 12 | \(\Z/30\Z\) | not in database |
| $18$ | 18.0.110625655867925924191778703000000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.359703397629795211516224000000000000.1 | \(\Z/6\Z\) | not in database |
| $20$ | 20.0.1959070718382489867508411407470703125.2 | \(\Z/5\Z\) | 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 | split | ord | add | ord | ord | ord | ord | ord | ord | add | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | - | 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.