Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+134943x-27860535\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+134943xz^2-27860535z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+174886101x-1300385779290\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(170, -85)$ | $0$ | $2$ |
Integral points
\( \left(170, -85\right) \)
Invariants
| Conductor: | $N$ | = | \( 41574 \) | = | $2 \cdot 3 \cdot 13^{2} \cdot 41$ |
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| Discriminant: | $\Delta$ | = | $-492857766955367424$ | = | $-1 \cdot 2^{10} \cdot 3^{3} \cdot 13^{9} \cdot 41^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{56300788871783}{102108404736} \) | = | $2^{-10} \cdot 3^{-3} \cdot 13^{-3} \cdot 41^{-2} \cdot 38327^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0796812408031663329810308073$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.79720656207239796495428708652$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9417230995548135$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.495379503642896$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.15434255757588438712087604759$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 120 $ = $ ( 2 \cdot 5 )\cdot3\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.6302767272765316136262814277 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 4.630276727 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.154343 \cdot 1.000000 \cdot 120}{2^2} \\ & \approx 4.630276727\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 725760 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $41$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12792 = 2^{3} \cdot 3 \cdot 13 \cdot 41 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10297 & 4 \\ 7802 & 9 \end{array}\right),\left(\begin{array}{rr} 5906 & 1 \\ 9839 & 0 \end{array}\right),\left(\begin{array}{rr} 1601 & 11194 \\ 11192 & 1599 \end{array}\right),\left(\begin{array}{rr} 6397 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 12789 & 4 \\ 12788 & 5 \end{array}\right),\left(\begin{array}{rr} 8530 & 1 \\ 8527 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[12792])$ is a degree-$443647682150400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12792\Z)$.
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 | $4$ | \( 507 = 3 \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 13858 = 2 \cdot 13^{2} \cdot 41 \) |
| $5$ | good | $2$ | \( 20787 = 3 \cdot 13^{2} \cdot 41 \) |
| $13$ | additive | $98$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 41574.t
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 3198.b2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.4195776.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.26776499624349696.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 13 | 41 |
|---|---|---|---|---|---|
| Reduction type | split | split | ord | add | split |
| $\lambda$-invariant(s) | 3 | 1 | 2 | - | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.