Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-55124182x+148812599764\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-55124182xz^2+148812599764z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-71440940547x+6944072268694014\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5084, -2542)$ | $0$ | $2$ |
Integral points
\( \left(5084, -2542\right) \)
Invariants
| Conductor: | $N$ | = | \( 451770 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $1150612794875440005120000$ | = | $2^{28} \cdot 3^{5} \cdot 5^{4} \cdot 11 \cdot 37^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{7220044159551112609}{448454983680000} \) | = | $2^{-28} \cdot 3^{-5} \cdot 5^{-4} \cdot 11^{-1} \cdot 37^{3} \cdot 52237^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3683759268259175265595137106$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5629169705038053043754658751$ |
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| $abc$ quality: | $Q$ | ≈ | $1.035653777739774$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.998791009424264$ | |||
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: | $\Omega$ | ≈ | $0.085324869411500109091824330180$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot1\cdot2^{2}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.68259895529200087273459464144 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.682598955 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.085325 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 0.682598955\end{aligned}$$
Modular invariants
Modular form 451770.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 108380160 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{28}$ | nonsplit multiplicative | 1 | 1 | 28 | 28 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 | 8.12.0.11 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9768 = 2^{3} \cdot 3 \cdot 11 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 2480 & 7659 \\ 1517 & 5810 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 9761 & 8 \\ 9760 & 9 \end{array}\right),\left(\begin{array}{rr} 8104 & 7659 \\ 6253 & 5810 \end{array}\right),\left(\begin{array}{rr} 371 & 6438 \\ 962 & 2147 \end{array}\right),\left(\begin{array}{rr} 7697 & 3996 \\ 5846 & 7031 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 9762 & 9763 \end{array}\right),\left(\begin{array}{rr} 5807 & 0 \\ 0 & 9767 \end{array}\right)$.
The torsion field $K:=\Q(E[9768])$ is a degree-$36944982835200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9768\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 45177 = 3 \cdot 11 \cdot 37^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 150590 = 2 \cdot 5 \cdot 11 \cdot 37^{2} \) |
| $5$ | split multiplicative | $6$ | \( 30118 = 2 \cdot 11 \cdot 37^{2} \) |
| $7$ | good | $2$ | \( 225885 = 3 \cdot 5 \cdot 11 \cdot 37^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 41070 = 2 \cdot 3 \cdot 5 \cdot 37^{2} \) |
| $37$ | additive | $686$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 451770.q
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 330.c3, its twist by $37$.
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 $0$.