Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-13692450x-19055637875\)
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(homogenize, simplify) |
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\(y^2z=x^3-13692450xz^2-19055637875z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13692450x-19055637875\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2395, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2395, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 412200 \) | = | $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 229$ |
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| Discriminant: | $\Delta$ | = | $7427977844238281250000$ | = | $2^{4} \cdot 3^{12} \cdot 5^{18} \cdot 229 $ |
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| j-invariant: | $j$ | = | \( \frac{1557756837810558976}{40757080078125} \) | = | $2^{11} \cdot 3^{-6} \cdot 5^{-12} \cdot 229^{-1} \cdot 91283^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9781767235693647140751929110$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3931025628316112446047799188$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9650103458151377$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.711070074239518$ | |||
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: | $\Omega$ | ≈ | $0.078609062429525789724000717294$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.6598524949258568601280516452 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 5.659852495 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.078609 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 5.659852495\end{aligned}$$
Modular invariants
Modular form 412200.2.a.bh
For more coefficients, see the Downloads section to the right.
| Modular degree: | 31408128 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $4$ | $I_{12}^{*}$ | additive | 1 | 2 | 18 | 12 |
| $229$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 27480 = 2^{3} \cdot 3 \cdot 5 \cdot 229 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 27473 & 8 \\ 27472 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 18319 & 0 \\ 0 & 27479 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 21983 & 18312 \\ 14652 & 18287 \end{array}\right),\left(\begin{array}{rr} 17179 & 17178 \\ 14898 & 10315 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 27474 & 27475 \end{array}\right),\left(\begin{array}{rr} 24049 & 24048 \\ 1158 & 24055 \end{array}\right),\left(\begin{array}{rr} 9208 & 3 \\ 17205 & 18322 \end{array}\right)$.
The torsion field $K:=\Q(E[27480])$ is a degree-$2018670634598400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/27480\Z)$.
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 | $2$ | \( 51525 = 3^{2} \cdot 5^{2} \cdot 229 \) |
| $3$ | additive | $2$ | \( 45800 = 2^{3} \cdot 5^{2} \cdot 229 \) |
| $5$ | additive | $18$ | \( 16488 = 2^{3} \cdot 3^{2} \cdot 229 \) |
| $229$ | split multiplicative | $230$ | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 412200.bh
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 27480.c3, its twist by $-15$.
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 $0$.