Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1609288x+790262281\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1609288xz^2+790262281z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2085637275x+36901761549750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 68450 \) | = | $2 \cdot 5^{2} \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-3512479453921000000$ | = | $-1 \cdot 2^{6} \cdot 5^{6} \cdot 37^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{8398297}{64} \) | = | $-1 \cdot 2^{-6} \cdot 37 \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3885038389983720378344469529$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.82349372564816111237799649440$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9122311702877938$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.895031771476672$ | |||
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: | $\Omega$ | ≈ | $0.25141340538448747667734099917$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ ( 2 \cdot 3 )\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $6.0339217292276994402561839800 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 6.033921729 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.251413 \cdot 1.000000 \cdot 6}{1^2} \\ & \approx 6.033921729\end{aligned}$$
Modular invariants
Modular form 68450.2.a.bo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2301696 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $37$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
| $3$ | 3B | 9.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6660 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 37 \), index $288$, genus $6$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3701 & 2700 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3331 & 2700 \\ 1665 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 12 & 433 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 6625 & 36 \\ 6624 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 27 \\ 4653 & 3808 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 6610 & 6637 \end{array}\right),\left(\begin{array}{rr} 2839 & 1305 \\ 195 & 2824 \end{array}\right),\left(\begin{array}{rr} 2663 & 0 \\ 0 & 6659 \end{array}\right)$.
The torsion field $K:=\Q(E[6660])$ is a degree-$1133539246080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6660\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 34225 = 5^{2} \cdot 37^{2} \) |
| $3$ | good | $2$ | \( 34225 = 5^{2} \cdot 37^{2} \) |
| $5$ | additive | $14$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $506$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 68450.bo
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 2738.c1, its twist by $185$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.5476.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.119946304.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.6325293375.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.3748322000.5 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.9372992211440206484080078125.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.1036577954647595315487384000000000.1 | \(\Z/6\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | add | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 2 | - | 0 | 0 | 0 | 0 | 0 | 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 $0$.