Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-10634x-403369\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-10634xz^2-403369z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13781691x-18612850410\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-57, 175)$ | $0.58651692422210049883160821982$ | $\infty$ |
| $(825, 23107)$ | $2.7634583445170047157579390696$ | $\infty$ |
| $(-71, 35)$ | $0$ | $2$ |
Integral points
\( \left(-71, 35\right) \), \( \left(-67, 151\right) \), \( \left(-67, -85\right) \), \( \left(-57, 175\right) \), \( \left(-57, -119\right) \), \( \left(-49, 79\right) \), \( \left(-49, -31\right) \), \( \left(125, 427\right) \), \( \left(125, -553\right) \), \( \left(127, 497\right) \), \( \left(127, -625\right) \), \( \left(237, 3115\right) \), \( \left(237, -3353\right) \), \( \left(825, 23107\right) \), \( \left(825, -23933\right) \), \( \left(1007, 31297\right) \), \( \left(1007, -32305\right) \), \( \left(275897, 144779747\right) \), \( \left(275897, -145055645\right) \)
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $8207864368704$ | = | $2^{6} \cdot 3^{2} \cdot 7^{7} \cdot 11^{3} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{849278123}{52416} \) | = | $2^{-6} \cdot 3^{-2} \cdot 7^{-1} \cdot 13^{-1} \cdot 947^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2297335682506975023856552616$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.34269532447655178618250700461$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9269344000478278$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.0226983843217434$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6182932248759132738383322092$ |
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| Real period: | $\Omega$ | ≈ | $0.47195458681662619114607712808$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2 \cdot 3 )\cdot2\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $18.330261847066972389940652157 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 18.330261847 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.471955 \cdot 1.618293 \cdot 96}{2^2} \\ & \approx 18.330261847\end{aligned}$$
Modular invariants
Modular form 462462.2.a.fz
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1327104 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $13$ | $1$ | $I_{1}$ | split 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 5009 & 3004 \\ 1000 & 7007 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1460 & 1 \\ 3639 & 0 \end{array}\right),\left(\begin{array}{rr} 8005 & 4 \\ 8004 & 5 \end{array}\right),\left(\begin{array}{rr} 6866 & 1 \\ 4575 & 0 \end{array}\right),\left(\begin{array}{rr} 1850 & 1 \\ 3079 & 0 \end{array}\right),\left(\begin{array}{rr} 4005 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[8008])$ is a degree-$89270570188800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8008\Z)$.
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 | $4$ | \( 7007 = 7^{2} \cdot 11 \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 77077 = 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $42$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35574 = 2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 462462fz
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 66066cf1, its twist by $-7$.
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.