Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+1088656x+20335725396\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+1088656xz^2+20335725396z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+88181109x+14824479270330\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2020, 175446)$ | $0$ | $4$ |
Integral points
\( \left(-2597, 0\right) \), \((2020,\pm 175446)\)
Invariants
| Conductor: | $N$ | = | \( 121296 \) | = | $2^{4} \cdot 3 \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-178726225938674510831616$ | = | $-1 \cdot 2^{13} \cdot 3^{20} \cdot 7 \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{740480746823}{927484650666} \) | = | $2^{-1} \cdot 3^{-20} \cdot 7^{-1} \cdot 19^{-1} \cdot 83^{3} \cdot 109^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1406684325104276503109053014$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.97530176236726211088915946400$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0514505724300596$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.210529356160168$ | |||
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: | $\Omega$ | ≈ | $0.079307732818886797271250952341$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ 2\cdot( 2^{2} \cdot 5 )\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.1723093127554718908500380937 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.172309313 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.079308 \cdot 1.000000 \cdot 160}{4^2} \\ & \approx 3.172309313\end{aligned}$$
Modular invariants
Modular form 121296.2.a.cf
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11059200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3185 & 8 \\ 3184 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 920 & 3 \\ 1373 & 2 \end{array}\right),\left(\begin{array}{rr} 2348 & 3191 \\ 2329 & 3186 \end{array}\right),\left(\begin{array}{rr} 1992 & 391 \\ 1967 & 1920 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3186 & 3187 \end{array}\right),\left(\begin{array}{rr} 2129 & 8 \\ 2132 & 33 \end{array}\right),\left(\begin{array}{rr} 400 & 2003 \\ 1201 & 1230 \end{array}\right)$.
The torsion field $K:=\Q(E[3192])$ is a degree-$381250437120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.
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 | $4$ | \( 2527 = 7 \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 40432 = 2^{4} \cdot 7 \cdot 19^{2} \) |
| $5$ | good | $2$ | \( 40432 = 2^{4} \cdot 7 \cdot 19^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 17328 = 2^{4} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 121296cs
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 798d4, its twist by $76$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-266}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.13827744.4 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 | 19 |
|---|---|---|---|---|---|
| Reduction type | add | split | ord | nonsplit | add |
| $\lambda$-invariant(s) | - | 1 | 4 | 0 | - |
| $\mu$-invariant(s) | - | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.