Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+708392x-34111104\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+708392xz^2-34111104z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+57379725x-24694855614\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(48, 0)$ | $0$ | $2$ |
Integral points
\( \left(48, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 435344 \) | = | $2^{4} \cdot 7 \cdot 13^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-23246716249293601792$ | = | $-1 \cdot 2^{10} \cdot 7^{5} \cdot 13^{6} \cdot 23^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{7953970437500}{4703287687} \) | = | $2^{2} \cdot 5^{6} \cdot 7^{-5} \cdot 23^{-4} \cdot 503^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4054136717002432987063518295$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.54531634250285383949858134083$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0516056711569626$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.006954239081704$ | |||
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: | $\Omega$ | ≈ | $0.12515610162567427745818541175$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ 2^{2}\cdot5\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.5031220325134855491637082351 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.503122033 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.125156 \cdot 1.000000 \cdot 80}{2^2} \\ & \approx 2.503122033\end{aligned}$$
Modular invariants
Modular form 435344.2.a.bv
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9031680 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $7$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $23$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 8.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 56.12.0.n.1, level \( 56 = 2^{3} \cdot 7 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 29 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 9 & 50 \\ 48 & 7 \end{array}\right),\left(\begin{array}{rr} 34 & 1 \\ 39 & 0 \end{array}\right),\left(\begin{array}{rr} 53 & 4 \\ 52 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[56])$ is a degree-$258048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56\Z)$.
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 | $2$ | \( 1183 = 7 \cdot 13^{2} \) |
| $5$ | good | $2$ | \( 62192 = 2^{4} \cdot 13^{2} \cdot 23 \) |
| $7$ | split multiplicative | $8$ | \( 62192 = 2^{4} \cdot 13^{2} \cdot 23 \) |
| $13$ | additive | $86$ | \( 2576 = 2^{4} \cdot 7 \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 18928 = 2^{4} \cdot 7 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 435344.bv
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1288.c2, its twist by $-52$.
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 $0$.