Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-234984x-43402887\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-234984xz^2-43402887z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-304539939x-2020437000162\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{161080372755087674431}{65331129960202500}, \frac{1981411752119549603873264915021}{16698607418819276908875000}\right) \) | $43.201886933815527921165116121$ | $\infty$ |
| \( \left(-\frac{1221}{4}, \frac{1221}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([41172070790032669795110106050:1981411752119549603873264915021:16698607418819276908875000]\) | $43.201886933815527921165116121$ | $\infty$ |
| \([-2442:1221:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{161107594059237758806}{1814753610005625}, \frac{2001997787514565938770819968046}{77308367679718874578125}\right) \) | $43.201886933815527921165116121$ | $\infty$ |
| \( \left(-10974, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 123969 \) | = | $3 \cdot 31^{2} \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $20281201280575803$ | = | $3^{12} \cdot 31^{6} \cdot 43 $ |
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| j-invariant: | $j$ | = | \( \frac{1616855892553}{22851963} \) | = | $3^{-12} \cdot 11^{6} \cdot 43^{-1} \cdot 97^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9338205163284492376768816221$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.21682691408587611471229945983$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0580583409717772$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.153846635737982$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $43.201886933815527921165116121$ |
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| Real period: | $\Omega$ | ≈ | $0.21702580163546576639652036836$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.3759241439760691235016280533 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.375924144 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.217026 \cdot 43.201887 \cdot 4}{2^2} \\ & \approx 9.375924144\end{aligned}$$
Modular invariants
Modular form 123969.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 864000 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $31$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $43$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 31992 = 2^{3} \cdot 3 \cdot 31 \cdot 43 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 4496 & 10323 \\ 6293 & 24770 \end{array}\right),\left(\begin{array}{rr} 31985 & 8 \\ 31984 & 9 \end{array}\right),\left(\begin{array}{rr} 17543 & 0 \\ 0 & 31991 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 31986 & 31987 \end{array}\right),\left(\begin{array}{rr} 4000 & 26195 \\ 7099 & 28800 \end{array}\right),\left(\begin{array}{rr} 14323 & 14322 \\ 14074 & 26971 \end{array}\right),\left(\begin{array}{rr} 21329 & 6200 \\ 24428 & 24801 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[31992])$ is a degree-$4576833463910400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/31992\Z)$.
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 | $2$ | \( 41323 = 31^{2} \cdot 43 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 41323 = 31^{2} \cdot 43 \) |
| $31$ | additive | $482$ | \( 129 = 3 \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 2883 = 3 \cdot 31^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 123969.c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 129.b1, its twist by $-31$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{43}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-31}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1333}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-31}, \sqrt{43})\) | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | ord | nonsplit | ord | ss | ss | ord | ord | ord | ord | ord | add | ord | ord | split | ord |
| $\lambda$-invariant(s) | 20 | 3 | 3 | 1,1 | 1,1 | 1 | 1 | 1 | 3 | 1 | - | 1 | 1 | 2 | 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.