Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-7900862855x+269922348939647\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-7900862855xz^2+269922348939647z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-126413805675x+17274903918331750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(30499, 7555950)$ | $5.1710237191901885297449210433$ | $\infty$ |
| $(-102621, 51310)$ | $0$ | $2$ |
| $(52899, -26450)$ | $0$ | $2$ |
Integral points
\( \left(-102621, 51310\right) \), \( \left(30499, 7555950\right) \), \( \left(30499, -7586450\right) \), \( \left(52899, -26450\right) \)
Invariants
| Conductor: | $N$ | = | \( 450450 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $90608954099570156250000000000$ | = | $2^{10} \cdot 3^{14} \cdot 5^{16} \cdot 7^{2} \cdot 11^{4} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{4788502600127122071579248929}{7954695558810000000000} \) | = | $2^{-10} \cdot 3^{-8} \cdot 5^{-10} \cdot 7^{-2} \cdot 11^{-4} \cdot 13^{-2} \cdot 27337^{3} \cdot 61657^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.4521864839739690877305790851$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.0981613834228640547325768000$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0049534086902208$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.144131353318383$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.1710237191901885297449210433$ |
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| Real period: | $\Omega$ | ≈ | $0.033916863993865593228064206944$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1280 $ = $ ( 2 \cdot 5 )\cdot2^{2}\cdot2^{2}\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.030792655426132126150203078 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.030792655 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.033917 \cdot 5.171024 \cdot 1280}{4^2} \\ & \approx 14.030792655\end{aligned}$$
Modular invariants
Modular form 450450.2.a.kc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 707788800 |
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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 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $5$ | $4$ | $I_{10}^{*}$ | additive | 1 | 2 | 16 | 10 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 8191 & 3642 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7279 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 5203 & 3642 \\ 2598 & 7279 \end{array}\right),\left(\begin{array}{rr} 4201 & 7284 \\ 1122 & 3649 \end{array}\right),\left(\begin{array}{rr} 2183 & 3636 \\ 726 & 7271 \end{array}\right),\left(\begin{array}{rr} 5461 & 3642 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10917 & 4 \\ 10916 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$38954430627840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $8$ | \( 50050 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| $5$ | additive | $14$ | \( 9009 = 3^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 64350 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 34650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 450450.kc
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 30030.bb3, its twist by $-15$.
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.