Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+281437679x+1346696656945\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+281437679xz^2+1346696656945z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+22796451972x+981810252268848\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 432960 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 41$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2210261250929615136000000000$ | = | $-1 \cdot 2^{14} \cdot 3^{21} \cdot 5^{9} \cdot 11^{5} \cdot 41 $ |
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| j-invariant: | $j$ | = | \( \frac{150470198145383828085247664}{134903640803809517578125} \) | = | $2^{4} \cdot 3^{-21} \cdot 5^{-9} \cdot 7^{3} \cdot 11^{-5} \cdot 41^{-1} \cdot 73^{3} \cdot 413069^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9342963897800784317329735124$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.1256246791268089040795360374$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0141527597528714$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.392025553294436$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.030146482292937022748548259928$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 20 $ = $ 2^{2}\cdot1\cdot1\cdot5\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.4117185834349618198838607942 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.411718583 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.030146 \cdot 1.000000 \cdot 20}{1^2} \\ & \approx 2.411718583\end{aligned}$$
Modular invariants
Modular form 432960.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 217728000 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $1$ | $I_{21}$ | nonsplit multiplicative | 1 | 1 | 21 | 21 |
| $5$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $11$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $41$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 54120 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 41 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 40589 & 54114 \\ 40587 & 54101 \end{array}\right),\left(\begin{array}{rr} 21121 & 6 \\ 9243 & 19 \end{array}\right),\left(\begin{array}{rr} 15786 & 11281 \\ 18043 & 2274 \end{array}\right),\left(\begin{array}{rr} 54115 & 6 \\ 54114 & 7 \end{array}\right),\left(\begin{array}{rr} 19681 & 6 \\ 4923 & 19 \end{array}\right),\left(\begin{array}{rr} 27065 & 54114 \\ 27066 & 54113 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 37883 & 54114 \\ 32469 & 54101 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 27059 & 0 \\ 0 & 54119 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[54120])$ is a degree-$80441612697600000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/54120\Z)$.
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 | $2$ | \( 6765 = 3 \cdot 5 \cdot 11 \cdot 41 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 28864 = 2^{6} \cdot 11 \cdot 41 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 7872 = 2^{6} \cdot 3 \cdot 41 \) |
| $7$ | good | $2$ | \( 144320 = 2^{6} \cdot 5 \cdot 11 \cdot 41 \) |
| $11$ | split multiplicative | $12$ | \( 39360 = 2^{6} \cdot 3 \cdot 5 \cdot 41 \) |
| $41$ | nonsplit multiplicative | $42$ | \( 10560 = 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 432960.t
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 27060.l2, its twist by $8$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.