Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-71551875x-232997818750\)
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(homogenize, simplify) |
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\(y^2z=x^3-71551875xz^2-232997818750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-71551875x-232997818750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 68400 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-7797929058720000000000$ | = | $-1 \cdot 2^{14} \cdot 3^{9} \cdot 5^{10} \cdot 19^{5} $ |
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| j-invariant: | $j$ | = | \( -\frac{1389310279182025}{267418692} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-3} \cdot 5^{2} \cdot 19^{-5} \cdot 31^{3} \cdot 1231^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2011185851111317202469502980$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.61746699985538125296479611373$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0156937260858383$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.916732007602206$ | |||
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: | $\Omega$ | ≈ | $0.025954298148399430968872440451$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot2^{2}\cdot1\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0381719259359772387548976180 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.038171926 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.025954 \cdot 1.000000 \cdot 40}{1^2} \\ & \approx 1.038171926\end{aligned}$$
Modular invariants
Modular form 68400.2.a.et
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6912000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $19$ | $5$ | $I_{5}$ | split 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 \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 567 & 1130 \\ 1120 & 731 \end{array}\right),\left(\begin{array}{rr} 781 & 10 \\ 485 & 51 \end{array}\right),\left(\begin{array}{rr} 1131 & 10 \\ 1130 & 11 \end{array}\right),\left(\begin{array}{rr} 569 & 1130 \\ 565 & 1089 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 379 & 1130 \\ 755 & 1089 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 1085 & 1021 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1140])$ is a degree-$5673369600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1140\Z)$.
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 | $2$ | \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \) |
| $3$ | additive | $6$ | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
| $5$ | additive | $2$ | \( 144 = 2^{4} \cdot 3^{2} \) |
| $19$ | split multiplicative | $20$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 68400fh
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 2850g1, its twist by $60$.
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 |
|---|---|---|---|
| $3$ | 3.1.5700.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.0.18000.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.7407720000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.2.3149280000000000.25 | \(\Z/5\Z\) | 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 | add | add | add | ord | ord | ord | ord | split | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | - | - | - | 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 $0$.