# Properties

 Label 245.c3 Conductor $245$ Discriminant $-5044200875$ j-invariant $$\frac{71991296}{42875}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, 425, 433])

gp: E = ellinit([0, -1, 1, 425, 433])

magma: E := EllipticCurve([0, -1, 1, 425, 433]);

$$y^2+y=x^3-x^2+425x+433$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(89, 857\right)$$ $\hat{h}(P)$ ≈ $0.12245926738832446151095000050$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1, 2\right)$$, $$\left(-1, -3\right)$$, $$\left(19, 122\right)$$, $$\left(19, -123\right)$$, $$\left(89, 857\right)$$, $$\left(89, -858\right)$$, $$\left(7399, 636402\right)$$, $$\left(7399, -636403\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$245$$ = $5 \cdot 7^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-5044200875$ = $-1 \cdot 5^{3} \cdot 7^{9}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{71991296}{42875}$$ = $2^{15} \cdot 5^{-3} \cdot 7^{-3} \cdot 13^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.55111105382460654116272165490\dots$ Stable Faltings height: $-0.42184402070305011138995471682\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.12245926738832446151095000050\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.83342839777879595302770655636\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $12$  = $3\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $1.2247323721513969719086898183907418856$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q - q^{3} - 2 q^{4} + q^{5} - 2 q^{9} - 3 q^{11} + 2 q^{12} - 5 q^{13} - q^{15} + 4 q^{16} - 3 q^{17} - 2 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 96 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $4$ $I_{3}^{*}$ Additive -1 2 9 3

## Galois representations

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

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
$3$ 3Cs 3.12.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss ordinary split add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 2,7 1 2 - 1 1 1 3 1 1 3 1 1 1 1 0,0 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.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 245.c consists of 2 curves linked by isogenies of degrees dividing 9.

## 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{21})$$ $$\Z/3\Z$$ 2.2.21.1-175.1-b3 $2$ $$\Q(\sqrt{-7})$$ $$\Z/3\Z$$ 2.0.7.1-175.1-a3 $3$ 3.1.140.1 $$\Z/2\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.686000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.3704400.1 $$\Z/6\Z$$ Not in database $6$ 6.0.137200.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.13722579360000.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ 12.0.343064484000000.3 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ 12.0.470596000000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.8838621364756451414483823486328125.2 $$\Z/9\Z$$ Not in database $18$ 18.0.1839302601739695422127.2 $$\Z/9\Z$$ 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.