Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+56133x+2362041\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+56133xz^2+2362041z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+898125x+152068750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 4950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-13758536601562500$ | = | $-1 \cdot 2^{2} \cdot 3^{7} \cdot 5^{10} \cdot 11^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{2747555975}{1932612} \) | = | $2^{-2} \cdot 3^{-1} \cdot 5^{2} \cdot 11^{-5} \cdot 479^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7851964893015870201314097171$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.10530791539421813773351234572$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9936024056521444$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.221498125472854$ | |||
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: | $\Omega$ | ≈ | $0.25139845548286514443843920724$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.0111876438629211555075136579 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.011187644 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.251398 \cdot 1.000000 \cdot 8}{1^2} \\ & \approx 2.011187644\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 48000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $11$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 327 & 650 \\ 640 & 131 \end{array}\right),\left(\begin{array}{rr} 651 & 10 \\ 650 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 654 & 655 \\ 445 & 4 \end{array}\right),\left(\begin{array}{rr} 606 & 5 \\ 595 & 656 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 325 & 656 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 605 & 541 \end{array}\right)$.
The torsion field $K:=\Q(E[660])$ is a degree-$608256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/660\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \) |
| $3$ | additive | $8$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
| $5$ | additive | $2$ | \( 18 = 2 \cdot 3^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 4950k
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 1650d1, its twist by $-15$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.3300.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\zeta_{15})^+\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.1437480000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.648402306750000.5 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.49207500000000.3 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | - | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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 $0$.