Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-2031605x-1115245897\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-2031605xz^2-1115245897z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-164560032x-812520578844\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-40351/49, 8450/343)$ | $2.2085251761785392683837461592$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 125060 \) | = | $2^{2} \cdot 5 \cdot 13^{2} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $714367732000000$ | = | $2^{8} \cdot 5^{6} \cdot 13^{6} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{750484394082304}{578125} \) | = | $2^{19} \cdot 5^{-6} \cdot 7^{6} \cdot 23^{3} \cdot 37^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1580611399502824477778134505$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.41348834084621720680624831541$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1088030644745788$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.702115930921964$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.2085251761785392683837461592$ |
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| Real period: | $\Omega$ | ≈ | $0.12645608688743087197041058732$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ 3\cdot( 2 \cdot 3 )\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.054132256588829693906162711 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.054132257 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.126456 \cdot 2.208525 \cdot 36}{1^2} \\ & \approx 10.054132257\end{aligned}$$
Modular invariants
Modular form 125060.2.a.j
For more coefficients, see the Downloads section to the right.
| Modular degree: | 933120 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $37$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2886 = 2 \cdot 3 \cdot 13 \cdot 37 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1964 & 39 \\ 2119 & 1561 \end{array}\right),\left(\begin{array}{rr} 2663 & 0 \\ 0 & 2885 \end{array}\right),\left(\begin{array}{rr} 2185 & 1560 \\ 117 & 1795 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 2881 & 6 \\ 2880 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2886])$ is a degree-$859600594944$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2886\Z)$.
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 | $2$ | \( 6253 = 13^{2} \cdot 37 \) |
| $3$ | good | $2$ | \( 25012 = 2^{2} \cdot 13^{2} \cdot 37 \) |
| $5$ | split multiplicative | $6$ | \( 25012 = 2^{2} \cdot 13^{2} \cdot 37 \) |
| $13$ | additive | $86$ | \( 740 = 2^{2} \cdot 5 \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 125060e
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 740b2, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.148.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.810448.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.48026889947088.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.1299323376.3 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.284038912193981413850629264444915071000000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.6.151655037446837122437969641679799388952711168.1 | \(\Z/6\Z\) | 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 | add | ord | split | ord | ord | add | ss | ord | ss | ss | ord | nonsplit | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 2 | 3 | 1 | - | 1,1 | 1 | 1,1 | 1,1 | 1 | 5 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 1 | 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.