Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-3877313x+1380301761\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-3877313xz^2+1380301761z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-314062380x+1005297796656\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2127, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2127, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 78144 \) | = | $2^{6} \cdot 3 \cdot 11 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $2909028302025840918528$ | = | $2^{42} \cdot 3 \cdot 11^{5} \cdot 37^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{24591016773082896625}{11097062309363712} \) | = | $2^{-24} \cdot 3^{-1} \cdot 5^{3} \cdot 11^{-5} \cdot 37^{-2} \cdot 581597^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8137390370395292012093275888$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7740182661996112370834794066$ |
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| $abc$ quality: | $Q$ | ≈ | $1.00898253428439$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.070477329552137$ | |||
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: | $\Omega$ | ≈ | $0.12820689719682024471228668347$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2^{2}\cdot1\cdot5\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.2820689719682024471228668347 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.282068972 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.128207 \cdot 1.000000 \cdot 40}{2^2} \\ & \approx 1.282068972\end{aligned}$$
Modular invariants
Modular form 78144.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4055040 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{32}^{*}$ | additive | 1 | 6 | 42 | 24 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $37$ | $2$ | $I_{2}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4884 = 2^{2} \cdot 3 \cdot 11 \cdot 37 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1630 & 1 \\ 1627 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3665 & 1222 \\ 1220 & 3663 \end{array}\right),\left(\begin{array}{rr} 4881 & 4 \\ 4880 & 5 \end{array}\right),\left(\begin{array}{rr} 2666 & 1 \\ 3551 & 0 \end{array}\right),\left(\begin{array}{rr} 3961 & 4 \\ 3038 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[4884])$ is a degree-$9236245708800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4884\Z)$.
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 | $2$ | \( 33 = 3 \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 26048 = 2^{6} \cdot 11 \cdot 37 \) |
| $5$ | good | $2$ | \( 7104 = 2^{6} \cdot 3 \cdot 37 \) |
| $11$ | split multiplicative | $12$ | \( 7104 = 2^{6} \cdot 3 \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 2112 = 2^{6} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 78144.bc
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2442.i2, its twist by $8$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.2891328.5 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.9103797810302976.110 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 11 | 37 |
|---|---|---|---|---|---|
| Reduction type | add | nonsplit | ss | split | nonsplit |
| $\lambda$-invariant(s) | - | 0 | 2,2 | 3 | 0 |
| $\mu$-invariant(s) | - | 0 | 0,0 | 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 $0$.