Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+60921x+27564705\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+60921xz^2+27564705z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+78953589x+1285822015686\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-168, 3633)$ | $0.60659196044723719280604980258$ | $\infty$ |
| $(-945/4, 945/8)$ | $0$ | $2$ |
Integral points
\( \left(-168, 3633\right) \), \( \left(-168, -3465\right) \), \( \left(756, 22113\right) \), \( \left(756, -22869\right) \), \( \left(3108, 172347\right) \), \( \left(3108, -175455\right) \)
Invariants
| Conductor: | $N$ | = | \( 134862 \) | = | $2 \cdot 3 \cdot 7 \cdot 13^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-342588705325310808$ | = | $-1 \cdot 2^{3} \cdot 3^{4} \cdot 7^{8} \cdot 13^{6} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( \frac{5180411077127}{70976229912} \) | = | $2^{-3} \cdot 3^{-4} \cdot 7^{-8} \cdot 11^{9} \cdot 13^{3} \cdot 19^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0453494524510472698298667059$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.76287477372027890180312298512$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1650936017955584$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.045618098980521$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.60659196044723719280604980258$ |
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| Real period: | $\Omega$ | ≈ | $0.22491355814579121957017093355$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 384 $ = $ 3\cdot2^{2}\cdot2^{3}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $13.097352592014640319904409432 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.097352592 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.224914 \cdot 0.606592 \cdot 384}{2^2} \\ & \approx 13.097352592\end{aligned}$$
Modular invariants
Modular form 134862.2.a.bp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1769472 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5928 = 2^{3} \cdot 3 \cdot 13 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 5471 & 0 \\ 0 & 5927 \end{array}\right),\left(\begin{array}{rr} 3953 & 2288 \\ 2132 & 3225 \end{array}\right),\left(\begin{array}{rr} 2224 & 1547 \\ 4849 & 3966 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 5922 & 5923 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1756 & 5473 \\ 4511 & 3654 \end{array}\right),\left(\begin{array}{rr} 859 & 1768 \\ 650 & 5877 \end{array}\right),\left(\begin{array}{rr} 5921 & 8 \\ 5920 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[5928])$ is a degree-$4956255682560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5928\Z)$.
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 | $4$ | \( 3211 = 13^{2} \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 22477 = 7 \cdot 13^{2} \cdot 19 \) |
| $7$ | split multiplicative | $8$ | \( 19266 = 2 \cdot 3 \cdot 13^{2} \cdot 19 \) |
| $13$ | additive | $86$ | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 134862c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 798b4, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-38}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{247}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-26}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-26}, \sqrt{-38})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | ord | split | ss | add | ord | split | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 2 | 3 | 2 | 1,1 | - | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.