Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-612033x-171936063\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-612033xz^2-171936063z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-49574700x-125490114000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-353, 0\right) \) | $0$ | $2$ |
| \( \left(897, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-353:0:1]\) | $0$ | $2$ |
| \([897:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-3180, 0\right) \) | $0$ | $2$ |
| \( \left(8070, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-543, 0\right) \), \( \left(-353, 0\right) \), \( \left(897, 0\right) \)
\([-543:0:1]\), \([-353:0:1]\), \([897:0:1]\)
\( \left(-543, 0\right) \), \( \left(-353, 0\right) \), \( \left(897, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 91200 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 19$ |
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| Minimal Discriminant: | $\Delta$ | = | $1871424000000000000$ | = | $2^{18} \cdot 3^{4} \cdot 5^{12} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{6189976379881}{456890625} \) | = | $3^{-4} \cdot 5^{-6} \cdot 7^{3} \cdot 19^{-2} \cdot 43^{3} \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2520049916038494961499279491$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.40756526454688134472370010030$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9556029786132598$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.516950051369824$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.17148490319654609955987562774$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.68593961278618439823950251095 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.685939613 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.171485 \cdot 1.000000 \cdot 64}{4^2} \\ & \approx 0.685939613\end{aligned}$$
Modular invariants
Modular form 91200.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1769472 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | 1 | 6 | 18 | 0 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $19$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 8.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 760 = 2^{3} \cdot 5 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 77 & 4 \\ 154 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 63 & 2 \\ 338 & 379 \end{array}\right),\left(\begin{array}{rr} 383 & 756 \\ 384 & 755 \end{array}\right),\left(\begin{array}{rr} 757 & 4 \\ 756 & 5 \end{array}\right),\left(\begin{array}{rr} 379 & 0 \\ 0 & 759 \end{array}\right),\left(\begin{array}{rr} 191 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[760])$ is a degree-$1891123200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/760\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 30400 = 2^{6} \cdot 5^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 3648 = 2^{6} \cdot 3 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 91200bl
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 285c2, its twist by $40$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-10}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-95})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | \(\Q(i, \sqrt{2}, \sqrt{5}, \sqrt{19})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 19 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | add | split |
| $\lambda$-invariant(s) | - | 0 | - | 1 |
| $\mu$-invariant(s) | - | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.