Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-40099398837x+3090699934894909\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-40099398837xz^2+3090699934894909z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3248051305824x+2253110508384471216\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1040692/9, 625807/27)$ | $1.1633975842777366901379489739$ | $\infty$ |
| $(118924, 1958887)$ | $1.2946555877732511883354168778$ | $\infty$ |
Integral points
\((106492,\pm 5301023)\), \((118924,\pm 1958887)\), \((817020,\pm 718127137)\)
Invariants
| Conductor: | $N$ | = | \( 435344 \) | = | $2^{4} \cdot 7 \cdot 13^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-1111491378430658671373398016$ | = | $-1 \cdot 2^{12} \cdot 7^{4} \cdot 13^{7} \cdot 23^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{360675992659311050823073792}{56219378022244619} \) | = | $-1 \cdot 2^{15} \cdot 7^{-4} \cdot 13^{-1} \cdot 23^{-9} \cdot 47^{3} \cdot 473287^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.5939423030476109265018809478$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.6183204437568972490579051056$ |
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| $abc$ quality: | $Q$ | ≈ | $1.03901187680286$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.535597040843599$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5020440351039714414642532521$ |
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| Real period: | $\Omega$ | ≈ | $0.038343991646730109331084367854$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 144 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot3^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $8.2935884066468354004465979259 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.293588407 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.038344 \cdot 1.502044 \cdot 144}{1^2} \\ & \approx 8.293588407\end{aligned}$$
Modular invariants
Modular form 435344.2.a.o
For more coefficients, see the Downloads section to the right.
| Modular degree: | 846526464 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $23$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
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 | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 75348 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \cdot 23 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 37673 & 0 \\ 0 & 75347 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 18 \\ 21708 & 463 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 8704 & 9 \\ 26073 & 37666 \end{array}\right),\left(\begin{array}{rr} 75347 & 75330 \\ 0 & 71161 \end{array}\right),\left(\begin{array}{rr} 37691 & 75330 \\ 37692 & 75329 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 75331 & 18 \\ 75330 & 19 \end{array}\right),\left(\begin{array}{rr} 13114 & 9 \\ 3267 & 37666 \end{array}\right)$.
The torsion field $K:=\Q(E[75348])$ is a degree-$36588435897581568$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/75348\Z)$.
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 | $2$ | \( 3887 = 13^{2} \cdot 23 \) |
| $3$ | good | $2$ | \( 18928 = 2^{4} \cdot 7 \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 62192 = 2^{4} \cdot 13^{2} \cdot 23 \) |
| $13$ | additive | $98$ | \( 2576 = 2^{4} \cdot 7 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 18928 = 2^{4} \cdot 7 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 435344.o
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 2093.h1, its twist by $-52$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.